yacht master water resistance

.css-1obzumv{font-weight:700;font-size:clamp(1.875rem, 1.25rem + 1.5625vw, 3.125rem);line-height:1.2;margin-bottom:1rem;line-height:1.1;}.css-1obzumv:lang(th){line-height:1.5;} Yacht-Master 42 .css-18uwo57{font-size:clamp(1.125rem, 1.0625rem + 0.1563vw, 1.25rem);line-height:1.6;font-weight:300;line-height:1.2;text-wrap:balance;}.css-18uwo57 span{display:block;} Oyster, 42 mm, RLX titanium Reference 226627

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Staying on course

The Oyster Perpetual Yacht-Master 42 in RLX titanium with an intense black dial and an Oyster bracelet.

Bidirectional rotatable bezel, timing the distance.

The Yacht-Master’s bidirectional rotatable 60-minute graduated bezel is made entirely from precious metals or fitted with a Cerachrom insert in high-tech ceramic. The raised polished numerals and graduations stand out clearly against a matt, sand-blasted background.

This functional bezel – which allows the wearer to calculate, for example, the sailing time between two buoys – is also a key component in the model’s distinctive visual identity.

Intense black dial

Exceptional legibility.

Like all Rolex Professional watches, the Yacht-Master 42 offers exceptional legibility in all circumstances, and especially in the dark, thanks to its Chromalight display.

The broad hands and hour markers in simple shapes – triangles, circles, rectangles – are filled with a luminescent material emitting a long-lasting glow.

RLX titanium

Ultralight resistance

RLX titanium is a grade 5 titanium alloy specially selected by Rolex. Like all titanium alloys, it is especially lightweight and is noted for its mechanical strength and corrosion resistance.

Another characteristic of RLX titanium is the possibility of working it to give a polished or satin finish according to the brand’s specifications. Its high mechanical strength makes it complex to work with, and the decision to use it has required the introduction of special production processes.

The Oyster bracelet

Alchemy of form and function.

The Yacht-Master 42, made from RLX titanium, is fitted on an Oyster bracelet. Developed at the end of the 1930s, this three-piece link bracelet remains the most universal in the Oyster Perpetual collection and is known for its robustness.

The Oyster bracelet of this new version of the Yacht-Master 42 features the Oysterlock folding safety clasp, which prevents accidental opening. It is also equipped with the Easylink comfort extension link, developed by Rolex, which allows the wearer to easily adjust the bracelet length by approximately 5 mm. The Oyster bracelet in RLX titanium also includes patented ceramic inserts – designed by the brand – inside the links to enhance its flexibility on the wrist and its longevity.

More Yacht-Master technical details

Reference   226627

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Oyster, 42 mm, RLX titanium

Oyster architecture

Monobloc middle case, screw-down case back and winding crown

Bidirectional rotatable 60-minute graduated bezel with matt black Cerachrom insert in ceramic, polished raised numerals and graduations

Winding crown

Screw-down, Triplock triple waterproofness system

Scratch-resistant sapphire, Cyclops lens over the date

Water resistance

Waterproof to 100 metres / 330 feet

Perpetual, mechanical, self-winding

3235, Manufacture Rolex

-2/+2 sec/day, after casing

Centre hour, minute and seconds hands. Instantaneous date with rapid setting. Stop-seconds for precise time setting

Paramagnetic blue Parachrom hairspring. High-performance Paraflex shock absorbers

Bidirectional self-winding via Perpetual rotor

Power reserve

Approximately 70 hours

Oyster, three-piece solid links

Folding Oysterlock safety clasp with Easylink 5 mm comfort extension link

Intense black

Highly legible Chromalight display with long-lasting blue luminescence

Certification

Superlative Chronometer (COSC + Rolex certification after casing)

Learn how to set the time and other functions of your Rolex watch by consulting our user guides.

Yacht-Master 42

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Rolesium Yacht-Master 116622

With its debut just two years ago at Baselworld 2016, the Yacht-Master ref. 116622 has quickly established itself as a coveted Rolex luxury watch. Part sporty part precious, this modern Rolesium Yacht-Master 40 plays up both sides beautifully. Let’s explore all the glorious details.

The Rolesium Yacht-Master 40

Whereas Rolesor refers to Rolex combining steel and gold on a two-tone watch, Rolesium is when rugged stainless steel and ultra-precious platinum meet on a Rolex watch. The Rolex ref. 116622 is indeed a Rolesium Yacht-Master model where the 40mm Oyster case and sporty Oyster bracelet are crafted in stainless steel while the bezel is made from platinum.

Although steel and platinum are both white metals – thus lend a monochromatic look to the watch – the Yacht-Master offers great contrasting textures. From the opposing high-polished raised numerals on the bezel set against a sandblasted background to the polished center links on the bracelet flanked by the brushed-finish outer links, the Yacht-Master ref. 116622 is always interesting to look at.

In terms of dial options, there’s the dark rhodium dial accented with a turquoise seconds hand and the turquoise YACHT-MASTER text or the blue dial with red accents. There’s plenty of lume on the Rolesium Yacht-Master for ideal legibility in low light and of course, the signature date window at 3 o’clock along with the Cyclops magnification lens on the sapphire crystal.

The Yacht-Master’s Caliber 3135

Often dubbed as Rolex’s workhorse, the Caliber 3135 has been an important movement for the company since 1988. This particular automatic movement powers many of Rolex’s timepieces including the Rolesium Yacht-Master ref. 116622.

The Caliber 3135 provides a power reserve of around 48 hours and as of 2015—a year before the launch of the Yacht-Master ref. 116622—Rolex redefined their “Superlative Chronometer Officially Certified” designation to guarantee an impressive accuracy rating of -2/+2 seconds per day. The self-winding caliber also boasts the paramagnetic blue Parachrom hairspring for improved resistance to magnetic fields and daily knocks.

Date Function

Naturally, as one of Rolex’s newer models, the Rolesium Yacht-Master 116622 includes the quickset date function where the date window is adjusted independently from the center timekeeping hands, in addition to the hacking feature where the seconds hands stops when the crown is pulled out for precise time-setting.

Ready For Marine Lifestyle

Finally, as its name suggests, the Yacht-Master 40 is perfectly suited for a marine lifestyle thanks in part to its 100-meter (330 feet) water resistance. To keep the water and dust out, there’s the Triplock screw-down winding crown and the fluted caseback.

A wonderful addition to the Rolex catalog, the   Rolesium Yacht-Master 40 is casually elegant, yet practical and durable. Just the thing to wear while sailing the seas.

What do you think about the The Rolesium Yacht-Master 40? Do you like Rolesium Rolex watches? Share your opinions with us in the comment section below.

About Paul Altieri

Paul Altieri is a vintage and pre-owned Rolex specialist, entrepreneur, and the founder and CEO of BobsWatches.com. - the largest and most trusted name in luxury watches. He is widely considered a pioneer in the industry for bringing transparency and innovation to a once-considered stagnant industry. His experience spans over 35 years and he has been published in numerous publications including Forbes, The NY Times, WatchPro, and Fortune Magazine. Paul is committed to staying up-to-date with the latest research and developments in the watch industry and e-commerce, and regularly engages with other professionals in the industry. He is a member of the IWJG, the AWCI and a graduate of the GIA. Alongside running the premier retailer of pre-owned Rolex watches, Paul is a prominent Rolex watch collector himself amassing one of the largest private collections of rare timepieces. In an interview with the WSJ lifestyle/fashion editor Christina Binkley, Paul opened his vault to display his extensive collection of vintage Rolex Submariners and Daytonas. Paul Altieri is a trusted and recognized authority in the watch industry with a proven track record of expertise, professionalism, and commitment to excellence.

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Rolex Yacht-Master Watch Review

Rolex yacht-master review.

Rolex has built a reputation on their classic and timeless designs. They’re not one to release a new model every other year. In fact, after the launch of the Daytona in 1963, it would be nearly another 30 years before they’d debut an entirely new model. That model was the Rolex Yacht-Master. Here, we’ll provide a Rolex Yacht-Master review, including key features of the Rolex Yacht-Master, the history of the Rolex Yacht-Master through the years, and notable wearers.

List of Key Features of the Rolex Yacht-Master

• Case: Available in three different sizes – 37mm, 40mm, and 42mm – and two materials – gold or two-tone gold and platinum
• Bezel: Bidirectional, rotatable, 60-minute graduated bezel
• Crown: Screw-down, Triplock, water-resistant crown system
• Depth Rating: 100 meters of water resistance
• Movement: Perpetual, mechanical, self-winding movement manufactured by Rolex
• Band: Offered with either an Oyster bracelet or Oysterflex strap

History of the Rolex Yacht-Master through the Years

The very first  Rolex Yacht-Master  was the Reference 16628. The brand only offered the model in a 40mm, 18-karat yellow gold case with a white dial. The Ref. 16628 also came equipped with a screw-down, Triplock crown and boasted 100 meters of water resistance. Inside, it housed the Caliber 3135 movement.

For the first several years, Rolex only made minor changes to the Yacht-Master line. In 1994, they released a slightly different iteration of the Yacht-Master: the Reference 68628. This variation was smaller, with a 35mm case. That same year, they also added a women’s version of the Yacht-Master: the Reference 69628. It showcased an even more modest case size, measuring just 29mm.

In 1999, Rolex introduced the first major update to the Yacht-Master collection. That year, they debuted an all-new, patented combination of metals created specifically for the Yacht-Master. They called this two-tone combination of stainless steel and platinum, Rolesium . At the annual Basel World Fair, Rolex launched the material in three different sizes. These included the 40mm Reference 16622, 35mm Reference 168622, and the 29mm Reference 169622.

The next update to the Yacht-Master collection came in 2005. That year, Rolex added another two-tone variation to the line, this time in stainless steel and 18-karat yellow gold. They offered the new colorway in the 40mm Reference 16623. Two years later, Rolex made the most significant change to the Yacht-Master line with the addition of the Yacht-Master II . However, the lineage of the original Yacht-Master has continued.

In 2012, Rolex released the next notable upgrade for the Yacht-Master with the Reference 116622. While the model retained its 40mm sizing, it featured an all-new “super case” with different styling. In addition, it boasted an all-platinum bezel as opposed to a combination of platinum and stainless steel. Last but not least, it came equipped with a refined version of the Oyster bracelet featuring an upgraded clasp.

Three years later, Rolex debuted another first for the brand in the Yacht-Master collection’s Reference 116655. This time, instead of a new metal, they introduced their own rigorously designed and tested variation of the rubber strap. The Oysterflex bracele t marked the first-ever rubber strap for the brand. Ever since, it’s become a staple of the Yacht-Master collection.

In the past several years, Rolex has continued to make subtle updated and additions to the Yacht-Master line. One of the latest releases is the Yacht-Master 40 with a multi-color, gem-set bezel. More recently in 2019, Rolex introduced the first 42mm time-and-date Yacht Master Reference 226659.

Deep Dive on Key Features of the Rolex Yacht-Master

For years, 40mm was the standard sizing for the Yacht-Master. The smaller, 35mm variation was the only alternative up until around 2016. Rolex has since replaced it with the 37mm iteration for a smaller option. In addition, it’s only been since 2019 that Rolex has made a larger, 42mm option available.

One of the most notable key features of the Rolex Yacht-Master is the bidirectional, rotatable, 60-minute graduated bezel. Its design helps skippers measure and anticipate the crucial countdown interval leading up to the start of a regatta or sailing race. The screw-down, Triplock crown is another key feature of the Rolex Yacht-Master. This water-resistant system has been a staple of the model since its inception.

With only 100 meters of water resistance, the Yacht-Master is perfect for enjoying a day on the water as opposed to scuba diving. Yet, its in-house, perpetual, mechanical, self-winding movement makes it a robust watch for any occasion. In addition, the option of Oyster bracelet or Oysterflex strap made it versatile enough to take from land to sea.

Who Wears the Rolex Yacht-Master?

The Yacht-Master is a popular choice among many of today’s top entertainers, athletes, and chefs. Some of the Yacht-Master’s famous wearers include TV personality Ellen DeGeneres and film icons like Brad Pitt and Bruce Willis . Athletes across an array of modalities also appreciate the Yacht-Master. You can find it on the wrist of star players like former pro-footballer David Beckham, Atlanta Falcons quarterback Matt Ryan , and former World Number One golfer Justin Thomas. Last but certainly not least, the legendary chef Emeril Lagasse is among the Yacht-Master’s celebrity fans.

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Rolex’s Titanium Yacht-Master 42 Review: The Game-Changer

Table of Contents

In the world of luxury watches , the new Rolex Yacht-Master 42 Ref. 226627 is a study in contrasts—nautical yet luxurious, familiar yet distinct. Introduced as part of a line that has often lived in the shadow of the iconic Submariner , this particular model breaks the mold with its RLX Titanium build and unique features.

It’s a distinct shift that has long-time fans and new admirers debating its merits. Is this the watch that finally elevates the Yacht-Master collection to legendary status?

If you’ve ever found yourself torn between the sporty appeal and luxury essence of Rolex watches, the Yacht-Master 42 Ref. 226627 demands your attention. Dive into our in-depth Yatch Master 42 review to explore its unique features and find out if this timepiece ticks all the right boxes for you.

The Oyster case of the Rolex Yacht-Master 42 Ref. 226627 stands out at 42mm, constructed entirely of RLX titanium. Notably, this isn’t any ordinary titanium. Rolex specifically opts for grade 5 titanium alloy for this model, prized for its extraordinary lightness paired with mechanical robustness and resistance to corrosion.

What sets RLX titanium apart further is its adaptability in the finish. Whether you’re looking at a polished gleam or a satin touch, it conforms to Rolex’s exacting standards. However, it’s worth noting that this high mechanical strength presents a challenge in crafting. To harness its qualities, Rolex had to innovate, integrating specialized production techniques tailored for RLX titanium.

Moreover, this monobloc middle case features a screw-down back and winding crown, embodying Rolex’s commitment to durability and precision engineering. The rotating bezel is bidirectional, complete with a Cerachrom insert matte black ceramic insert bezel, graced with a mirror polish, raised numerals and graduations. This perfectly complements the titanium case, making it an exemplar of maritime luxury.

Function meets form in its winding crown, equipped with Rolex’s Triplock triple waterproofness system. Coupled with a scratch-resistant sapphire crystal and a Cyclops lens over the date, this Titanium YM is waterproof to 100 meters or 330 feet and with a power reserve of up to 70 hours. 

The Dial 

The dial of the Rolex Yacht-Master 42 is more than a display, it’s a testament to horological engineering and aesthetic acumen. Enhanced with Chromalight technology, the matte black dial promises unimpeachable legibility even in the darkest marine abyss. Broad, polished numerals and geometrically distinct hands and markers comprising circles, rectangles, and triangles are filled with luminescent material that emits an enduring glow.

Such meticulous attention to detail reaffirms the watch’s utility, making it an indomitable tool for nocturnal navigation. Paired with its stunning deep black hue, this display adds a layer of mystique that’s equally at home on the high seas or at a black-tie affair.

The Strap/Bracelet

Adorning the new Yacht-Master 42 is an Oyster bracelet meticulously forged from RLX titanium, an alchemy of strength and refinement. A legacy creation tracing its roots to the late 1930s, this tri-link wonder stands as an iconic feature within Rolex’s revered Oyster Perpetual series.

The bracelet is enhanced with the Oysterlock folding safety clasp, an ingenious feature that secures your timepiece against accidental openings. It is also equipped with Rolex’s Easylink comfort extension link, offering the wearer the luxury of micro-adjustments for an impeccable fit.

Not to be overlooked, patented ceramic inserts embellish the internal links of this RLX titanium composition. These delicate yet durable additions not only enhance the bracelet’s ergonomic comfort but also significantly elevate its lifespan, epitomizing enduring elegance in horological artistry.

The Movement

The movement within the Rolex Yacht-Master 42 Ref. 226627 is a perpetual, mechanical, self-winding system that utilizes Rolex’s own Calibre 3235. This isn’t just any in-house movement; it’s one that assures a precision rate of -2/+2 seconds per day after casing, a feat that outperforms many other luxury timepieces in the market.

When it comes to functionality, this watch features a straightforward, yet highly useful set of capabilities. The central hour, minute, and seconds hands are complemented by an instantaneous date display function with rapid setting and a stop-seconds mechanism for ultra-precise time adjustment.

Underpinning this all is the oscillator, which employs a paramagnetic blue Parachrom hairspring and high-performance Paraflex shock absorbers. These features enhance the watch’s durability and performance under different conditions. The bidirectional self-winding Perpetual rotor ensures that the watch remains operational and accurate, even when not manually wound for an extended period.

The Competitive Landscape

These watches are competitors primarily due to their shared characteristics and target market. While not all of them are made of titanium, they share key features such as high water resistance, durable sapphire crystals, and reliable movements:

• ORIS PROPILOT X CALIBRE 115: Crafted from both Grade 2 and 5 Titanium, this timepiece offers a commendable 100m water resistance and houses a mechanical movement. Its impeccable titanium construction and precision mechanics align harmoniously with the Yacht-Master’s ethos of opulence and artistry.
• OMEGA SEAMASTER DIVER 300: While its case boasts Grade 2 Titanium, the Seamaster is celebrated for its extraordinary 300m water resistance, automatic movement, and the elegance of a sapphire crystal. It stands as a formidable choice for those who seek a timepiece of enduring strength and style.
• TUDOR PELAGOS 39: Enveloped in Grade 2 Titanium, this horological masterpiece showcases a robust 200m water resistance and houses an automatic movement. The Pelagos stands as a worthy contender in the realm of luxury sports watches, boasting durability and performance akin to the Yacht-Master.
• TAG HEUER AQUARACER PROFESSIONAL 300: Boasting a formidable 300m water-resistant case, automatic movement, and the refined allure of a sapphire crystal, the Aquaracer commands attention within the upper echelons of dive watches, despite the absence of titanium in its construction.
• VACHERON CONSTANTIN OVERSEAS TOURBILLON SKELETON: While featuring Grade 5 titanium, this watch stands out as a luxury option with a tourbillon movement. Its water resistance is lower at 50m, making it more of a statement piece than a diving watch , but it competes in the luxury segment.

Notable People Wearing Rolex Yacht Master 42

The Rolex Yacht-Master 42mm in RLX titanium has already made its way onto some high-profile wrists, creating buzz and drawing attention to this unique piece.

Sir Ben Ainslie

Image Source: Rolex Magazine

First seen sported by British competitive sailor Sir Ben Ainslie, the Yacht-Master 42’s maritime features and innovative titanium build found a perfect match. As a sailor with numerous accolades, Sir Ben Ainslie’s endorsement brings credibility to the Yacht-Master’s nautical roots and functionality.

Image Source: Time and Tide

Hollywood superstar Tom Cruise has also been spotted wearing the Rolex Yacht-Master 42. Known for his love of action and adventure, both on-screen and off, Cruise’s choice in wearing this Rolex model underscores its appeal to those who lead high-paced, adventurous lives.

The appearance of this Rolex timepiece on the wrists of individuals of such varied but high-profile backgrounds speaks to the universal appeal and versatility of the Yacht-Master 42. 

Why You Should Invest?

Priced at SGD 18,132.17, the Yacht-Master 42 Ref. 226627 represents a new frontier for Rolex, which has historically been associated with weighty, robust deepsea challenge timepieces made from stainless steel or precious metals. When a prototype was seen on the wrist of British competitive sailor Sir Ben Ainslie, the notion of a titanium Rolex suddenly moved from fantasy to reality.

Despite its lighter weight, this doesn’t undermine the watch’s value; it enhances it. The use of grade 5 RLX titanium introduces a level of complexity and craftsmanship that justifies its price tag. Titanium is notoriously difficult to work with, and Rolex has managed to not only manufacture it but also perfect it, maintaining the watch’s renowned durability and resistance.

When looking at Rolex watches as an investment, consider their volatility. The Yacht-Master series exhibits lower volatility, with other Yacht masters like the Rolex 226659 at 7.2% and Rolex 226658 at 4.4%, suggesting more stable value retention. Given the groundbreaking nature of the Titanium Yacht-Master Ref. 226627, its value is likely to appreciate over time. Rolex’s commitment to innovation while maintaining quality will likely make this model a sought-after piece in the future.

Pricing and Availability

The Rolex Yacht-Master 42 Ref. 226627 is priced at SGD 18,132.17 . Given its high-quality craftsmanship, innovative use of RLX titanium, and the brand’s reputation for durability and luxury, this price point is aligned with what one would expect for a timepiece of this caliber.

As for availability, it’s advisable to consult authorized Rolex dealers or reputable online platforms for the most current information. Rolex watches, especially innovative or popular models like this one, can sometimes be difficult to find in stock due to high demand and limited releases.

If you’re considering this tool watch as an investment piece or a functional yet luxurious watch, it’s recommended to act promptly, given the notable personalities already spotted wearing it and its potential for future value appreciation.

The Rolex watch Yacht-Master 42 Ref. 226627 is a groundbreaking entry in Rolex’s esteemed line-up, blending innovation with classic craftsmanship. Its use of RLX titanium and a host of advanced features mark it as a pinnacle of both style and functionality. 

Key Takeaways

• Introduced as a trailblazing model featuring RLX Titanium, this watch has redefined what it means to be a Rolex Yacht-Master.
• Its unique material and craftsmanship not only justify its premium price but also suggest a strong potential for value appreciation.
• The likes of Sir Ben Ainslie and Tom Cruise being spotted with this timepiece not only elevates its status but also underscores its universal appeal and functionality.

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Any other year, the titanium yacht-master 42 would steal the show for rolex. this year, the brand has so many crazy releases that the ym flies under the radar. here's why it still matters..

A year ago, the very idea of a titanium Rolex was relegated to wild dreams. A prototype had been seen on the wrist of British competitive sailor Sir Ben Ainslie, but the widely circulated online photo had gotten so old that some of us began to wonder if the watch would ever see the light of day.

The pic that launched our titanium dreams. Image by Ineos Britannia Team / C GREGORY

Now, in less than five months, we've gotten two watches from the Crown cased in RLX Titanium (a grade 5 titanium). The first was last year's 50mm Deep Sea Special , the mega dive watch that obliterated the water-resistance record. And now this week we have the Yacht-Master 42, which unlike the DSS is sized so that a normal human being could conceivably wear it. 

Here it is. The first practically sized titanium Rolex, the new Yacht-Master 42. 

It's a big deal. But when seen next to Daytonas with display casebacks, Day-Dates with emojis, a solid-gold GMT-Master II, and an entirely new line of dress watches, a titanium Yacht-Master barely moves the needle of surprise and excitement. What a wild 48 hours this has been for the House of Wilsdorf.

In some ways, it feels like the appropriate response to not be that excited. After all, at this point every other watchmaker under the sun has made a titanium watch, from affordable Citizens in multiple colors of bezels and dials to Jean-Claude Biver's $500,000 minute repeater tourbillon announced Sunday.

And yet, as soon as the new titanium Yacht-Master ref. 226627 started to be passed around the room of Hodinkee editors during this week's Watches & Wonders trade show, the general reaction was just to laugh with surprise. This 42mm watch, which looks so sturdy, feels so unbelievably light. I mean, that's titanium for you. But still. You can't quite believe this watch is real, on a number of different levels. 

For any of us who've ever tried on a steel Submariner (a.k.a. anyone with a passing interest in Rolex), it's kind of comical to find out how much your brain is preconditioned to see a 42mm steel Oyster case, round indices, and Mercedes hands and think about the luxurious heft that awaits you.

At around 100 grams, according to Rolex, the titanium Yacht-Master is so light it breaks your brain.

For a moment, let's compare the new YM to last year's titanium Pelagos from Rolex's sister brand Tudor. Rolex's choice to put the watch on a bracelet instead of a sportier Oysterflex makes the comparison obvious. I've now spent time with both pieces, and I prefer the Yacht-Master. 

The YM, like the Pelagos, is distinctly a tool watch – something that would have been hard to say about Yacht-Masters in the past. But the finishing a world apart, which is saying something for such an understated metal as titanium.

Rolex's proprietary grade 5 "RLX Titanium" (stronger than the grade 2 of the Pelagos) has the curious property of being equally able to be brushed satin or polished, which means it has the nice sharp and shiny chamfers that you'd like to see contrasted against the dark grey and relatively matte metal. That combination also works well with the more matte and textured dial – and with the contrast of the raised black numerals against a matte ceramic bezel insert, which is is the main giveaway that this is still squarely a Yacht-Master.

My main critique (which I share into the void, knowing that Rolex designers will do whatever they think best) is that I wish they'd  stuck to the no-date design of Ainsilie's prototype. In the practical application of most sailing races, there's really no use for a date. If you're blue-water sailing and circumnavigating the globe, maybe its useful, though just like dive watches the practical application gives way to the reality of technology. So why not refine the design further and leave the date off altogether? And while we're at it, a better quick-adjustment option would be great.

The price is somewhat immaterial – CHF 13,400 – since the average collector won't be able to get it at retail anytime soon. But the new Yacht-Master 42 is more than a solid release. It's a more than a titanium proof of concept. It's a wearable piece that portends at least the possibility of future experiments with this fascinating material. 

For more information visit Rolex. 

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SwissWatchExpo Ultimate Guide to the Rolex Yacht-Master and Yacht-Master II

The Rolex Yacht-Master collection pays homage to the brand’s connection to the world of sailing. Precision and dependability are vital for tools used in navigating open seas, and the Yacht-Master’s reliable, waterproof, and robust qualities make it the perfect timepiece for the sport.

While the original Rolex Yacht-Master is a stylish and ultra-luxe sport accessory for boating enthusiasts, the Yacht-Master II put together the best of Rolex technology to accurately time regatta races. Whether you’re lounging on a boat, or racing one, there is a Yacht-Master watch for you.

Learn more with SwissWatchExpo ‘s Ultimate Guide to the Rolex Yacht-Master collection.

Rolex Yacht-Master II Rolesor Everose Gold Steel 116681

About the Rolex Yacht-Master Collection

The Rolex Yacht-Master is one of the brand’s youngest collections, having been introduced only 1992. And yet, it’s already one of their most diverse, with two distinct models under the family of watches, and dozens of permutations when it comes to dial designs, bezel styles, and bracelet options.

A look into its history shows that Rolex had plans for a yacht-themed watch decades before the Yacht-Master line was launched. Today, it’s a mainstay in the Rolex catalog.

Here are important milestones in the Rolex Yacht-Master and Yacht-Master II history.

1958: Rolex entered a partnership with the prestigious New York Yacht Club, a social and boating club which sponsors an annual regatta.

1966: Rolex solidified its ties to the world of sailing when Francis Chichester – one of the world’s pioneering aviators and solo sailors – became the first person to sail around the globe while wearing a Rolex on his wrist. He sailed on his yacht, the Gipsy Moth IV, from August 1966 to May 1967, taking him 29,600 miles around the world.

1970s: Rolex never got around to creating a watch specifically for this category. The brand did make a prototype of the Cosmograph chronograph with the name “Yacht Master” on its dial, but they ultimately decided not to bring it to market.

1992: Rolex finally introduced the modern Yacht-Master – the brand’s first luxury sports watch built for the open seas. Whereas Rolex’s previous sports watches had utilitarian themes – the Submariner for diving, the Daytona for racing, and so on – the Yacht-Master was focused on aesthetics and luxury.

1994:  Rolex introduces midsize and ladies’ models of the Yacht-Master

1996: Rolex introduces two-tone (steel and gold) midsize and ladies’ models of the Yacht-Master

1997:  Rolex introduces the Rolesium version, which combines a steel body with a platinum bezel and dial. Until today the Rolesium is found exclusively in the Rolex Yacht-Master collection.

2007:  Rolex releases the Yacht-Master II, which is the world’s first watch equipped with a programmable countdown timer and a mechanical memory. The Yacht-Master II was unlike anything Rolex had launched before. Not only was it massive with its 44mm case, but it also had a bold, visually striking design that stood out among Rolex’s catalog of versatile timepieces.

2019:  Rolex introduces the Yacht-Master 42 to the collection. This is the first 42mm Yacht-Master model, offering a larger option for those who find 40mm too small. The first model came in 18k white gold, also a first for the collection, followed by an 18k gold version in 2022.

Key Features of the Rolex Yacht-Master

The Rolex Yacht-Master collection is composed of two families of watches: the original Yacht-Master, and the Yacht-Master II. While these two models have the same name, they are quite different from each other.

The Yacht-Master was introduced in 1992 as an ultra-luxurious sports watch for yachting and sailing enthusiasts. The nautical-inspired timepiece takes after the Rolex Submariner, but with greater emphasis on luxury materials and aesthetics.

The Yacht-Master II is a relatively new addition to the Rolex portfolio, introduced in 2007. Much larger that the Yacht-Master at 44mm, it is a purpose-built regatta watch equipped with a countdown timer for use in competitive sailing.

FEATURES OF THE ROLEX YACHTMASTER

Rolex yacht-master design.

Rolex Yacht-Master models come in an Oyster case with 100 meters of water resistance, a Triplock winding crown, and a bi-directional rotating bezel with raised 60-minute graduations and numerals. To make the watch easily readable, the dial features lume-filled round, baton, and triangular hour markers; luminous Mercedes-style hands, a date window at 3 o’clock, and a sapphire crystal with a Cyclops lens for easier reading of the date.

Since the introduction of the first all-gold Rolex Yacht-Master model in 1992, Rolex has expanded the line in terms of materials, aesthetic details, and mechanical upgrades. Here are the options available within the collection.

ROLEX YACHT-MASTER SIZES

The Yacht-Master line is one of the most varied Rolex collections due mainly to the number of sizes made available. The smaller sizes, particularly the Lady Yacht-Master models, have been discontinued, but are still available in the pre-owned market.

29mm (discontinued)

35mm (discontinued)

ROLEX YACHT-MASTER MATERIALS

The solid 18k yellow gold and two-tone models of the Yacht-Master have been discontinued and replaced with the Everose and White Gold versions. The 37 and 40mm sizes come in Everose, Everose Rolesor, and Rolesium metals, while the 42mm comes exclusively in White Gold.

Yellow Gold (discontinued)

Yellow Rolesor (Steel and Yellow Gold – discontinued)

Everose Gold

Everose Rolesor (Steel and Everose Gold)

Rolesium (Oystersteel and platinum)

ROLEX YACHT-MASTER BEZELS

The bezel material used for the Yacht-Master depends on the metal or alloy used for the case.

Matte Cerachrom bezel – 18k Everose and white gold models

18k Gold bezel – Rolesor steel and gold models

950 Platinum – Rolesium steel and platinum models

ROLEX YACHT-MASTER BRACELETS

The Rolex Yacht-Master has only been equipped with the three-link Oyster bracelet and the Oysterflex bracelet, which combines rubber with an internal flexible metal blade. The Oysterflex can only be found on the solid 18k Everose and white gold models with matte ceramic bezels.

ROLEX YACHT-MASTER MOVEMENTS

The Yacht-Master uses a time-and-date movement that changes depending on the size of the watch. Below are the movements used for each size.

29mm – Caliber 2135; Caliber 2235

35mm – Caliber 2135; Caliber 2235

37mm – Caliber 2236

40mm – Caliber 3135; Caliber 3235

42mm – Caliber 3235

FEATURES OF THE ROLEX YACHTMASTER II

Rolex yacht-master ii design.

The Rolex Yacht-Master II is one of the brand’s largest watches. Each model comes with a 44mm Oyster case that is water resistant to 100 meters, a Triplock winding crown, a Ring Command Bezel, two chronograph pushers, and an Oyster bracelet.

It is also one of Rolex’s most complicated watches, as it features a regatta chronograph with a countdown that can be programmed from 1 to 10 minutes, allowing yacht racers to accurately time the start of a race. Furthermore, its Ring Command Bezel is more than just a decorative element – it operates in conjunction with the movement and is actually how the user can set the programmable countdown.

Rolex Yacht-Master II Rolesium and Everose Rolesor Models

ROLEX YACHT-MASTER II MATERIALS

All Rolex Yacht-Master II models come in the same 44mm case size. Models are available in the following metals:

Yellow Gold

White Gold and Platinum

Everose Rolesor (stainless steel and Everose gold)

Oystersteel

The materials used for the bezel differ depending on the material of the case. Below are the available configurations.

Cerachrom Bezel – Yellow Gold, Everose Rolesor, Oystersteel

950 Platinum – White Gold

ROLEX YACHT-MASTER II DIALS

The Yacht-Master II dial is unlike anything seen on other Rolex models because of its niche complication. It features a countdown display which can be programmed from 1 to 10 minutes, and read using the red-tipped arrow hand. Below the hour and minute hands at the center is a running seconds sub-dial. There’s also a red central chronograph hand, which can fly back or fly-forward to its starting position while in motion. It also features 12 square lume-filled hour markers.

ROLEX YACHT-MASTER II MOVEMENTS

The Yacht-Master II is equipped with two of Rolex’s most advanced movements to date, which feature a programmable countdown timer with mechanical memory, and chronograph synchronization.

Earlier models were powered by the 4160, and later models were upgraded to the 4161. The latter is based on the Daytona’s Caliber 4130, which features the same blue Parachrom hairspring for better switch resistance, and offer a 72-hour power reserve.

Ref 116689 – Caliber 4160; Caliber 4161

Ref 116688 – Caliber 4160; Caliber 4161

Ref 116681 – Caliber 4160; Caliber 4161

Ref 116680 – Caliber 4161 .

Popular Rolex Yacht-Master Models

The Rolex Yacht-Master and Yacht-Master II are both excellent luxury watches that can suit any lifestyle. The Yacht-Master fits the mold of classic Rolex sports watches; while the Yacht-Master II has pushed the boundaries of what regatta watches can do. You can’t go wrong with the Rolex Yacht-Master, but here are some collector’s favorites:

ROLEX YACHT-MASTER with OYSTERFLEX BRACELET

In 2015, the Oysterflex, Rolex’s patented rubber bracelet, made its debut in the Yacht-Master collection. It’s appearance was also marked by several firsts: the first solid Everose gold Yacht-Master, paired with a matte dial and matte Cerachrom bezel.

The practical and edgy combination of solid gold and rubber continued with the release of the 18k white gold Yacht-Master 42 in 2019, and the 18k yellow gold version in 2022. This is a watch for someone who wants a luxurious watch that’s also ready for any adventure.

These models have the Calibre 3235 in place, which offers a 70-hour power reserve, and an upgraded Parachrom hairspring that has 10 times more precision than the traditional hairspring. The Calibre 3235 was also the first movement to receive Rolex’s certification as a Superlative Chronometer, and is regulated to a precision of an impressive -2/+2 seconds a day.

ROLEX YACHT-MASTER RHODIUM DIAL

The stunning combination of Oystersteel and 950 platinum was made even more beautiful with the addition of a sunburst rhodium dial. This luxurious watch is part of the series of Yachtmasters introduced in 2016, which were upgraded with the Caliber 3235 movement mentioned previously. The slate dial is paired with turquoise details on the model signature and seconds hand, which gives just the right amount of color on the sleek monochromatic design.

ROLEX YACHT-MASTER II STEEL EVEROSE GOLD

The highly specialized Yacht-Master II was created as a professional regatta watch which can also be used as a chronograph. On the bezel, it has large 0-10 scale numbers and the Yacht-Master II name. On the dial is a smaller version of this countdown. Together, these allow the wearer to select a countdown interval of anywhere from one to ten minutes.

Aside from being one of the most complicated watches in the Rolex portfolio, the Yacht-Master II also stood out because of its unique look, owing to its 44mm case and bold colors. The steel and 18k Everose gold version is a crowd favorite due to its eye-catching combination of rose gold, blue, and red hues.

Common Questions About the Rolex Yacht-Master

Built for the sea, but stylish enough for any occasion, the Rolex Yacht-Master is one of the most desired sports watches in the brand’s catalog. Here are some of the most common questions asked about the Rolex Yacht-Master.

WHAT IS A ROLEX YACHT-MASTER?

The Rolex Yacht-Master is a collection of watches built for yachting and sailing. It is composed of two families of watches: the Yacht-Master, a luxury watch designed to be worn on while lounging on a boat; and the Yacht-Master II, a functional sports watch designed to accurately time the start of a regatta race.

WHY IS THE ROLEX YACHT-MASTER SO EXPENSIVE?

As a line of sport watches with special focus on aesthetics, the Rolex Yacht-Master collection evokes an unmistakable sense of luxury than can only be achieved by using the finest materials.

The Rolex Yacht-Master II in particular, is equipped with exclusive functions and features such as the programmable countdown with mechanical memory and the Ring Command Bezel.

The architecture, manufacturing, and innovative features of the Yacht-Master collection make it unique in the world of watches and therefore fetches a higher price tag.

Rolex Yacht-Master Mother of Pearl Dials

HOW WILL I KNOW IF MY ROLEX YACHT-MASTER IS GENUINE?

A genuine Rolex watch will be beautifully finished and decorated in keeping with the brand’s uncompromising quality standards. If you see any sign of imperfection on the watch, such as misaligned or misspelled text, unfinished surfaces, or wrong engravings and markings, it is likely a fake.

It is always advisable to do research beforehand so you can check if the details on your watch is in tune with the model’s features.

We also advise clients to only purchase a watch from a reputable and trusted dealer, who can provide a guarantee of authenticity on the watch.

IS THE ROLEX YACHT-MASTER WATERPROOF?

Rolex Yacht-Master watches are guaranteed water resistant to a depth of 100 meters of 330 feet. The collection was not made specifically for diving, but they come with screw-down crowns and case backs, which protect the movement from moisture and dust that can be encountered with sporting use.

The Rolex Yacht-Master and Yacht-Master II serve as the embodiment of the brand’s relationship with the world of sailing. Explore our collection of Rolex Yacht-Master watches at SwissWatchExpo.com .

WATCH REVIEW

Swanky sailor: reviewing the rolex oyster perpetual yacht-master.

In this feature from the WatchTime archives, we take a close look at the modern version of Rolex ’s Oyster Perpetual Yacht-Master , with black Cerachrom bezel and Oysterflex bracelet. Original photos are by Nik Schölzel.

A water-resistant Oyster case, large hour markers and bold hands are essential elements of Rolex’s Submariner , introduced in 1953 and made for use underwater. In contrast, Rolex’s Yacht-Master, launched in 1992, is a luxury liner – equally at home on board a yacht on the high seas or on land at a ritzy yacht club. But to enjoy this luxury you’ll need to pay almost $25,000 for the 40-mm Everose gold and Cerachrom ceramic version shown here. Stainless-steel versions of the Yacht-Master are priced about $13,000 less.

The Yacht-Master was the first watch in Rolex’s Professional Oyster Collection to come in three different case sizes: 29, 35 and 40 mm. The model we tested, launched last year, is offered in two sizes: 37 and 40 mm. We chose the larger version, which we measured at precisely 40.19 mm in diameter and 11.49 mm in height (excluding the magnifying “Cyclops” lens for the date).

The well-known Cyclops date lens was patented by Rolex in 1953 and introduced in 1954 on the Datejust. This magnifying device is made of sapphire, like the watch’s crystal, and has nonreflective coating on both sides. The jumping date advances exactly at midnight.

The watch is powered by a seasoned caliber, the Rolex 3135, used in the very first Yacht-Master in 1992. The 3135 debuted in 1988 in the Submariner. The blue Parachrom balance spring was added to the movement in 2005, five years after it was first introduced in the Cosmograph Daytona. Its paramagnetic alloy resists changes caused by temperature variations and magnetic fields.

The Parachrom balance spring is thinner than a human hair and up to 10 times more resistant to shocks than a conventional balance spring. Provided with an overcoil, it is attached to a large balance wheel with a variable moment of inertia. Fine adjustments are made using four gold Microstella regulating screws. The balance wheel is supported by a height-adjustable bridge. The entire construction ensures rate results that bring the Yacht-Master (as well as the other watches in the Oyster collection) to the rank of “Superlative Chronometer Officially Certified.” These words on the Yacht-Master’s dial mean that the watch has endured 15 days and nights of testing by COSC in addition to a series of tests conducted by Rolex in its own laboratory. Acceptable rate results for a Superlative Chronometer allow deviations of less than +/-2 seconds per day, while COSC’s test permits average deviations between -4 and +6 seconds per day. In addition, Rolex’s tests are carried out under conditions that correspond more closely to real-life situations than COSC’s tests and are conducted on fully assembled watches, while COSC tests just the movements.

We, too, tested the finished watch – fully wound and after running for 24 hours – first in five positions on the timing machine and then with a two-week wearing test on the wrist (or simulated on a winding machine). In our tests, the Yacht-Master remained within the standards specified by Rolex for a Superlative Chronometer. On the wrist it showed virtually no deviations over the two-week period. On the timing machine it showed a gain of about 1 second per day with minimal variations among the positions.

The tests performed by Rolex include a water-resistance check. Unlike the Submariner, the Yacht-Master’s water resistance is “only” 100 meters. This means that the Yacht-Master is not designed to be a professional dive watch , which requires water resistance of at least 200 meters.

The Yacht-Master’s bidirectional bezel also keeps it from being a dive watch. The bezel on a dive watch usually rotates in only one direction so it won’t show a shorter dive time if it’s repositioned inadvertently. But the Yacht-Master’s bezel is impressive: it has 120 ratchets and shows graduations in 5-minute increments using both Arabic numerals and line markers. The first quarter has well-defined minutes markers. Polished, raised graduations on the ceramic inlay stand in relief against a sandblasted, matte black background. The bezel is made of Cerachrom, Rolex’s ceramic material. The Ceramic inlay is set in a deeply grooved ring made of Everose gold, Rolex’s rose-gold alloy. A grooved caseback seals the case hermetically and can only be opened using a special tool. Three dots on the screw-down crown indicate that this watch has been sealed with the Triplock sealing system, a triple water- resistance system developed by Rolex.

The screw-down crown sits securely inside the case between two crown guards. Releasing the crown allows it to spring away from the midsection, which makes it easy to use for manual winding, rapid date change and setting the hands.

The dial has a characteristic Rolex look. The applied markers and elongated triangle at 12 o’clock are filled with Rolex’s luminous substance, Chromalight, and are displayed on a matte black background. The hour hand has a “Mercedes” circle filled with Chromalight and the seconds hand has a luminous Chromalight dot. The stark contrast of black and white ensures excellent legibility during the day; at night the Chromalight emits a blue glow for easy reading in the dark.

This watch is the first Rolex with an Oysterflex bracelet. (We use the term “bracelet” rather than “strap” because the Oysterflex, unlike standard rubber straps, has metal on the inside.) Combining a rubber strap with a gold case is nothing new, but at Rolex, known for its conservative approach to design, it’s a major innovation. The patented bracelet has a core made of nickel-titanium alloy blades, which provide excellent flexibility and are coated with a black elastomer, a synthetic type of rubber. When the material is subjected to tension and pressure, it returns to its original shape quickly. It resists environmental changes and is long lasting, waterproof and hypoallergenic – a good alternative to a metal bracelet. The bracelet’s black color goes well with the ceramic bezel and with the Everose gold case, presenting a modern, two-tone look.

Inside the bracelet is a patented cushioning system that increases wearing comfort. The bracelet accommodates changes in wrist size. The single-sided Oysterlock folding clasp made of Everose gold offers additional flexibility; it allows for three length adjustments. Screws attach the clasp securely to the bracelet. A safety bar makes opening the clasp more difficult but prevents it from opening accidentally.

SPECS: Manufacturer: Rolex SA, Rue François-Dussaud 3-7, 1211 Geneva, Switzerland Reference number: 116655 Functions: Hours, minutes, central seconds, date, bidirectional rotating bezel Movement: In-house Caliber 3135, automatic, “Superlative Chronometer” certified, 28,800 vph, 31 jewels, Kif shock absorbers, Glucydur balance with Microstella regulating screws, Parachrom balance spring with overcoil, 48-hour power reserve, diameter = 28.5 mm, height = 6.0 mm Case: Everose-gold Oyster case with black Cerachrom ceramic graduated bezel, sapphire crystal with Cyclops magnifying lens, water resistant to 100 m Bracelet and clasp: Oysterflex bracelet with Everose-gold, single-sided Oysterlock folding clasp Rate results: Deviations in seconds per 24 hours (Fully wound / after 24 hours) Dial up +3.1 / +2.9 Dial down +2.1 / +3.2 Crown up -0.7 / -1.6 Crown down -0.8 / -2.6 Crown left +2.5 / +2.6 Greatest deviation of rate 3.9 / 5.8 Average deviation +1.2 / +0.9 Average amplitude: Flat positions 292° / 265° Hanging positions 247° / 233° Dimensions: Diameter = 40 mm, height = 11 mm, weight = 154 g Variations: 37-mm case (Ref. 268655, with Caliber 2236, $22,000) Price: $24,950

gold on rubber = cool if one can afford it. I still prefer my S/S Y-M with PT dial & bezel = an understated goes everywhere quite comfortable piece. By the way… hardly mentioned BUT the curved lugs on all Yacht-Master models make for a wonderfully contoured fit + way better than straight lugs on most Rolex’

You’ll have to sell it to pay for the repairs for your yacht… but you’ll sure look good wearing it!

Buy rolex oys

An interesting piece, shame about the poor out of focus photo’s !!!

May I request for an article or review regarding swiss made SELECTRON chronograph / computer watches made possibly during the 60s-70s?

That patent on the Cyclops would have expired over 40 years ago, meaning whatever it covered now can be freely-used by the public, I’m not sure it’s a bragging point (and, no, patents cannot be renewed).

This was a very nice watch. Is it automatic or manual? Do you know a good place to buy it?

Just aquired the YM 40 in Everose, stylish a touch sporty with a different look from my other Rolex’s.

Hey GW? Was wondering, since the article didn’t mention it, how is the rubber bracelet sized?

It actually comes with three different sizes, so you just pick the pair that works best for your wrist.

Sizing is a bit of a problem, I bought mine 2nd hand but never worn. AD will have various sizes in combination to make it fit. So if i were to sell it on that person may have a heck of a time getting it to fit.

Love the watch but I was told I had to wait a year to purchase the 40mm size in Canada. So, off I went and purchased a watch from a different manufacturerer!

Why make the consumer with cash in hand wait?

There are certainly other options available these days!

Old Caliber, a braclet which cannot be changed. What else can you expect from Rolex for a price about 24k Euro?

Yes, I agree. I don’t understand why they didn’t use the 3235 caliber.

Since spring 2019 they do use the 3235 movement.

Superb review of a superb Rolex. Thank you!

Hi, So does this mean that they have dumped the YM2 with its funny 10 second / minute countdown dial? I never liked the look of the YM2.

If they did it would explain why they are now available at Costco.

Truly lavishly stylish

Oh my goodness! Another lust object from Rolex (the origin of my watch, sorry chronometer, fetish). Faultless style and flash balanced with flawless functionality (OK, you can’t go all the way down). Opening my savings account now.

Rolex Oyster Perpetual Yacht-Master is truly a fabulous watch and though it cost me a lot of money, I must say it is worth the money that I had to pay for it.

Over the years it has been quite accurate and did not cause me any difficulty till date.

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THE ROLEX YACHT-MASTER: THE WATCH OF THE SEASON

The pedigree Rolex boasts of today as a watchmaker is a product of their distinctive and evergreen designs. The Rolex Yacht-Master is another great example. 

Rolex is not a brand that continues to introduce fresh models of wristwatches year after year. As a matter of fact, this watchmaking giant takes its time after every release before sending out another. 

For instance, the Daytona came into the scene in 1963. Thereafter, we had to wait for another 30 years before Rolex launched another new model. This model released in 1992 was the Rolex Yacht-Master. 

In this post, you get an extensive Rolex Yacht-Master review. Check out every detail of this amazing timepiece from Rolex with an entirely new and distinct outlook and function. 

• What are the different key specs of the Rolex Yacht-Master? 
• How did the Rolex Yacht-Master come to be through the years? 
• Who are the notable wearers of this wristwatch?
• Who is the wristwatch best for? 
• Is the Rolex Yacht-Master best for any particular season?

Get answers to all of these questions and more as you read on. Keep reading!

An Overview of The Important Features of the Rolex Yacht-Master

• Case Measurement And Variety: There are three distinct size varieties of the Rolex Yacht-Master. It can either be 37mm, 40mm, or 42mm. Also, it comes in two material varieties. Such as platinum, and gold, or two-tone gold.
• Bezel Details: The bezel is a 60-minute graduated, rotatable, bidirectional bezel.
• Crown Details: It features a Triplock, water-resistant, screw-down crown system
• Depth Rating: Rolex Yacht-Master can go as deep as 100 meters underwater.
• Movement Details: The timepiece features a self-winding, perpetual, mechanical movement built in-house by Rolex.
• Band Details: You either get the Oyster bracelet band or the Oysterflex strap band. 

As mentioned above, the Rolex Yacht-Master is not just any random release from Rolex. With the space of 30 years, the matchmaking world knew how much of a bombshell Rolex would drop in 1992.

Simply put, the Rolex Yacht-Master maintains a rather special spot in Rolex’s array of sports watches. The water-resistance feature combined with a rotatable bezel is not an entirely unique blend of features. These are two features Rolex Yacht-Master shares with the Submariner. 

However, the Submariner is water-resistant to as low as 300 m underwater. While the Yacht-Master water resistance is only as deep as 100 m underwater. You cannot refer to the Rolex Yacht-Master as a tool watch.

Furthermore, the Yacht-Master comes in different case materials varieties. There is the platinum and steel type. Also, there is the gold and steel type also referred to as Rolesium and Rolesor. Depending on the size and metal you opt for, you can either get something subtly or unequally luxurious.

How Did The Rolex Yacht-Master Come To Be Through The Years?

The Reference 16628 was the very first Rolex Yacht-Master to hit the world of wristwatch lovers. Rolex simply presented this model with a 40mm measurement. In addition, the case was an 18-karat yellow gold type. Plus, it came with a white dial. 

More so, the crown of this first version of the Yacht-Master was a Triplock crown that also featured a screw-down. Lastly, Ref. 16628 again had a water resistance feature as deep as 100 m below water level.  Plus, it came with the Caliber 3135 movement housed in its shell. 

As the years passed, Rolex continued to make alterations to the Yacht-Master line. Check out the different changes and the years they came into the picture. 

1994: The First Two Modifications

Initially, years into the production of this wristwatch, Rolex barely made any modifications to the Yacht-Master model. However, in 1994, the brand introduced a negligibly varied iteration of the Yacht-Master. That was the Reference 68628 model.

This new reference was a smaller variation as it came with a 35mm case. In the same year, Rolex again introduced a women’s edition of the Yacht-Master. This was the Reference 69628. This women’s edition presented an even more subtle case size. Its case measurement was a mere 29mm.

1999: Yet Another Update

There was another update in 1999. This time, Rolex made the first huge update to the Yacht-Master lineup. 

That year, Rolex showcased a completely fresh, patented blend of metals made particularly specific to the Yacht-Master timepiece. Rolex named this two-tone blend of platinum and stainless steel, Rolesium. 

The watchmaking brand unveiled the material used in 3 distinct sizes at the annual BaselWorld Fair. These distinct variations were:

• The 29mm Reference 169622.
• The 35mm Reference 168622
• The 40mm Reference 16622

2005: The Stainless Steel Addition

In 2005, we saw another update to the Yacht-Master line. In that year, Rolex introduced another two-tone variety to the collection. However, this time it came in stainless steel and 18-karat yellow gold case. Rolex presented the new colorway housed in the 40mm Reference 16623. 

Yet, barely two years later, Rolex gave us the most notable modification to the Yacht-Master collection. That year, we witnessed the launch of the Yacht-Master II. Nevertheless, this was not the end for the original Yacht-Master collection. The evolution continued. 

2012: The Reference 116622

Rolex debuted another new and significant upgrade to the Yacht-Master line in 2012. This upgrade came with the Reference 116622. 

For this variation, the brand maintained its 40mm size measurement. But, there was a completely fresh “super case”. This case came with a unique design. 

More so, it showed off an all-platinum bezel. This is a different feature compared to the usual blend of stainless steel and platinum. To wrap it up, this variation came furnished with a sophisticated version of the Oyster bracelet. This Oyster bracelet featured a refined clasp.

2015: Welcome The Rubber Strap

Only three years after Reference 116622, Rolex introduced yet another first of its kind. The brand launched the Yacht-Master Reference 116655 wristwatch. 

And, this time, Rolex didn’t go for new metal. Instead, they debuted their own precisely built and tested edition of the rubber strap. Plus, it was the Oysterflex bracelet that captioned the introduction of the first-ever rubber strap for the brand. Thereon, the rubber strap is now a staple of the Yacht-Master lineup.

Conclusively, a lot happened in the last few years. Rolex proceeds to create subtle upgrades and improvements to the Yacht-Master collection. A notable example of these additions is the latest release of the Yacht-Master 40. 

This addition presents a gem-set and multi-colored bezel. Most recently, in 2019, Rolex launched a maiden addition. This was the 42mm sized time-and-date Yacht-Master Reference 226659. We must add that this was yet another big one. 

Finally, the name “Yacht-Master” is moreover, incidentally, the term used for a certificate of competency in yachting. This term goes to a competitive athlete in yachting. The Royal Yachting Association is responsible for presenting this certificate of proficiency. 

It is not clear whether or not there is any particular link between the RYA and the Yacht-Master wristwatch. However, you will agree that this is one interesting discovery.

A Deeper Look Into The Major Features of the Rolex Yacht-Master

In this section, you get to discover a deeper view of the details of the Rolex Yacht-Master. You will find out more about the case sizes, the bezel, the water resistance, and so on. Follow through!

The Rolex Yacht-Master Case

For several years, the Rolex Yacht-Master maintained 40mm as the standard sizing. Hence, most of the Yacht-Master watches made had this sizing. Thereafter, there came a smaller version in the 35mm variation. This was the only other option up until another introduction in 2016. 

Over time, Rolex ensured there is a replacement for this with the 37mm iteration. With this iteration, we had another smaller alternative. 

Additionally, there was another introduction to this lineup in 2019. That year, Rolex built a bigger, 42mm alternative. This alternative is available to those who prefer a bigger wristwatch.

The Rolex Yacht-Master Bezel

The bezel is another perk that sets the Rolex Yacht-Master aside from the rest of the pack. It is one of the most significant and major features of the Rolex Yacht-Master. 

This is the rotatable, bidirectional, 60-minute graduated bezel. Thanks to its design, it is a tool watch for skippers. 

With this bezel, skippers can calculate and predict the important countdown period. So, they get ready for the beginning of a sailing race or regatta. 

The Rolex Yacht-Master Crown

More so, there is the screw-down and Triplock crown. This is also another major part of the Rolex Yacht-Master. The combination of the Triplock with the screw-down leaves barely anything more to be desired of the crown part of this timepiece. 

The Rolex Yacht-Master Water Resistance

In addition, there is the water-resistant system that comes with the Rolex Yacht-Master wristwatch. Over time, this feature evolved into a staple of this line of wristwatches. This is one partnership that existed from the launch of this timepiece. 

This Yacht-Master wrist watch comes with only 100 meters of water resistance. It is safe to say that the Yacht-Master wristwatch is ideal for basking a day on the water. It might not be the best watch to wear to scuba diving. 

The Rolex Yacht-Master Movement

This will be incomplete if we fail to add information about the movement housed in the Yacht-Master wristwatch. This is the in-house, self-winding, mechanical, perpetual movement. This movement goes on to make it a powerful timepiece. With the addition of this movement, the Rolex Yacht-Master will pass as the perfect timepiece for any outing. 

The Rolex Yacht-Master Oysterflex Strap

Finally, there is the presence of the Oysterflex strap. Thanks to this addition, this timepiece is now versatile enough. It can adapt to both the usual land conditions as well as the sea climate. Learn more about this innovative addition below. 

Simply put, the Oysterflex bracelet is rather an excellent display of effort. The Oysterflex strap is one of the most charming qualities of Rolex as a watchmaker  

This innovation is a rather clear demonstration of what one can only characterize as a remarkable level of utmost devotion. Rolex displays how much work the brand is willing to put into each of their creations with this one.

For this particular inclusion, we can say one thing for sure. The brand dug deep into research and dedication to give what is today a revolutionary innovation.

This Oysterflex comes with a simple design. The design of this addition features the hypoallergenic and relaxing elements of a rubber strap. Similarly, there are the strength and shape-retention details of a bracelet. 

Right in the heart of the Oysterflex bracelet, you have metal inserts. These inserts are products of nickel and titanium. With the help of these metal inserts, the bracelet gets fixed to the clasp and watch case firmly.

On top of that, there is a cover of “high-performance black elastomer.” For starters, “Elastomer” is a portmanteau term. It comes from the blend of the words “elastic” and “polymer”. This is a broad term to describe natural and man-made rubbers. 

To add to the sophistication of the materials of the Oysterflex bracelet, the bracelet comes shaped in a somewhat rare fashion. You have ridges chipped into the face part of the bracelet. With this input, the bracelet can now appear better on the wrist. This is possible as it blends seamlessly with the natural curvature of the wrist. This is always a perfect sight on the wrist when worn.

The Celebrity Fan Base Of The Rolex Yacht-Master ?

The Rolex Yacht-Master is a famous timepiece among several celebrities today. Most of the a-list athletes, entertainers, and chefs love this wristwatch.

From a long list of celebrity followings, here are some of the Yacht-Master’s popular wearers.

• On-screen icon Ellen DeGeneres 
• Movie stars like Bruce Willis and Brad Pitt 

The Yacht-Master wristwatch is also a beloved timepiece amongst athletes all over the world. Some of the star players that rocked this timepiece are: 

• Atlanta Falcons quarterback Matt Ryan
• Former pro-footballer David Beckham
• Former World Number One golfer Justin Thomas. 

Finally, the legendary chef, Emeril Lagasse appears on the list of admirers the Yacht-Master wristwatch have 

Conclusion  

In conclusion, it is not limited to how many productions a watchmaker has in its collection. A parade featuring well-executed and widely accepted timepieces might be all that is necessary. The Daytona and the Rolex Yacht-Master appearing on your collection is enough to boost the pedigree of your brand. Little wonder Rolex boasts of this Yacht-Master wristwatch as one of its best products. 

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cal yacht club rowing

California Yacht Club (CYC)

by evandiaz | May 19, 2023 | Yacht Club Members

California Yacht Club (CYC) calyachtclub.com 4469 Admiralty Way Marina del Rey, CA 90292 (310) 823-4567 Fax:(310) 822-3658

Total: 950 members – 33% power, 63% sail, 4% rowers.

Facilities: Bar open Wed. thru Sun 1000 to 2400. Dining room open Wed.- Sun for lunch and dinner (breakfast on Sat. & brunch on Sun.) Snack bar open every day from Memorial Day to Labor Day 1000 to 1630. Heated Pool and Paddle Tennis Courts available to members and their guests. 320 member slips (25’ to 120’), 100+ dry storage slips (up to 25’), two 2-ton hoists and launch ramp. Guest docks are first come, first served, but call for reservations. First night free for reciprocal club members. CYC monitors channel 68.

Junior program: Year round. Junior Sailing and Rowing Program. Open to all youths in the community (ages 8-18). Full time Junior staff.

Comments: Active power, sail and rowing fleets. Family friendly Club with a full schedule of races, cruises, and social activities.

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The 10 Best Rowing Clubs in Los Angeles

Los Angeles is a city with a rich history in rowing. Rowing clubs have been around in Los Angeles since the late 1800s, and many of them are still going strong today. In this blog post, we will take a look at the top 10 rowing clubs in Los Angeles and what makes them so successful. We will also discuss the different types of rowing that these clubs offer and how you can get involved. So if you’re looking for a place to row in Los Angeles , be sure to check out one of these great clubs!

Table of Contents

1: Los Angeles Rowing Club

The Los Angeles Rowing Club was founded in 1994 and is located in Marina del Rey, California. They are a non-profit organization that is open to the public. The LARC was created with the mission to promote the sport of rowing, and they do so by providing quality rowing programs for all levels of experience. They offer both sweep rowing and sculling programs, as well as learn-to-row classes. The LARC is also home to several competitive teams that compete at the local, state, and national level.

Some of their most notable achievements include winning the US Rowing Club National Championship in 2004 and sending two athletes to the Olympic Games in 2008. In addition to their competitive teams, the LARC also has a strong community outreach program that provides free rowing lessons to underserved youth in the Los Angeles area. The LARC is truly a place for everyone, whether you’re a seasoned rower or just looking to try something new.

2: Lions Rowing Club

The Lions Rowing Club in Los Angeles, USA has a beautiful and vastly successful history. The club was established in 1887, making it one of the clubs that paved the way for the newer clubs to arrive in the 1900s. In its early years, the club was based out of a boathouse on the Los Angeles River, but today it is located on Grand Canal in Long Beach. The Lions Rowing Club has produced many national champions and Olympic champions over the years, including Olympic gold medalist John Larissa and world champion rower Sarah Trowbridge.

The club is also linked to several colleges and universities in the area, including UCLA, USC, and Pepperdine. As a result of its successes both on and off the water, the Lions Rowing Club is widely regarded as one of the premier rowing clubs in the United States.

3: Long Beach Rowing Association

The Long Beach Rowing Association Rowing Club in Los Angeles, USA has an extremely interesting history. Founded in 1872, the club started to attract rowers in vast numbers. Throughout its history, the club has been dedicated to promoting the sport of rowing and developing its members into successful athletes. Today, the club is home to a diverse group of rowers of all ages and abilities. The club offers a variety of programs for both beginners and experienced rowers, making it one of the most inclusive rowing clubs in the area.

In addition to its excellent programs, the club is also linked to several colleges and universities, making it a great place for college rowers to train and compete. The Long Beach Rowing Association Rowing Club has a proud tradition of excellence and is committed to continuing to develop world-class rowers.

4: California Yacht Club

The California Yacht Club Rowing Club in Los Angeles is a rowing club with an impressive pedigree. Founded in 1994, the club has been home to many famous and successful rowers over the years. The club is located in Marina del Rey, just a short drive from downtown Los Angeles, and it remains one of the premier rowing clubs in the country.

The club is closely linked to UCLA, which is just down the road, and many of its members are student-athletes who row for the university. In recent years, the club has produced several Olympians and national champions, cementing its place as a top rowing destination in the United States.

The Importance of Hydration When Rowing

5: Metropolitan Rowing Club

The Metropolitan Rowing Club in Los Angeles has an interesting and storied history. The club was originally founded in 1884, making it one of the first rowing clubs to welcome people through their doors. In its early years, the club was based out of a boathouse on the banks of the Los Angeles River. The club quickly gained a reputation for excellence, and in 1886 they won their first National Championship. In the years since, the club has produced numerous national champions and Olympic rowers.

Today, the Metropolitan Rowing Club is based out of a state-of-the-art facility in Marina del Rey. The club is open to rowers of all ages and abilities, and they offer a variety of programs for both recreational and competitive rowers. Whether you’re looking to get fit, compete at the highest level, or just enjoy a day on the water, the Metropolitan Rowing Club is the perfect place for you.

6: Whitehall Spirit

The Whitehall Spirit Club in Los Angeles was founded in 1997. The club has produced many national and international champions, including two Olympic gold medalists. The Whitehall Spirit Club is also linked to several colleges and universities, making it a great place for students to learn and compete. If you’re looking for a top-notch rowing club with a rich history and plenty of success, the Whitehall Spirit Club is the perfect choice.

The Row LA Rowing Club in Los Angeles is a historic rowing club that was founded in 1985. The club is based out of Marina del Rey and is one of the most well-known rowing clubs in Southern California. The club is also linked to several colleges, including the University of Southern California and UCLA. The Row LA Rowing Club is a respected and prestigious rowing club that has a rich history and tradition of success.

8: Rivanna Rowing Club

Rivanna Rowing Club is located in Los Angeles, USA. The club has produced several successful rowers, including Olympians and national champions. The club is also linked to a college nearby, which provides access to facilities and equipment. The club has a strong focus on developing young rowers and providing them with opportunities to compete at the highest level.

The club has a rich history and tradition of winning, and this is reflected in the success of its members. Rivanna Rowing Club is a highly respected and well-known club, and it continues to produce champions and contribute to the sport of rowing.

9: Open Water Rowing Center

The Open Water Rowing Center is located in Los Angeles, USA and is a rowing club that is open to the public. The club has produced several famous and successful rowers, such as Olympian rower Sue Enquist and national champion rower George Plimpton. The club is also linked to several colleges nearby, such as the University of Southern California and UCLA.

The club has been very successful in producing Olympic champions and national champions. In addition, the club has also been successful in developing young athletes into professional rowers. The Open Water Rowing Center is an excellent place for anyone interested in learning how to row or for anyone looking to improve their rowing skills.

10: Duluth Rowing Club

Founded in 1966, the Duluth Rowing Club is a popular club in LA. Located in Los Angeles, the club has produced several national and international champions, including two Olympic gold medalists. The Duluth Rowing Club is also affiliated with the University of California, Los Angeles, one of the top rowing programs in the country. As a result, the Duluth Rowing Club is considered one of the premier rowing clubs in the United States.

The Duluth club has produced numerous national champions and Olympic medalists. In addition to its competitive success, the Duluth Rowing Club is also known for its beautiful rowing facility, which overlooks downtown Los Angeles. The club is open to rowers of all levels of experience, from beginners to experienced athletes. Whether you’re looking to compete at the highest level or simply enjoy a leisurely row on a beautiful day, the Duluth Rowing Club is the perfect place for you.

Application

Cyc jr. rowing inquiry form.

We are looking forward to hearing from you

CYC Rowing Participant Information

California Yacht Club is located at: 4469 Admiralty Way, Marina del Rey, CA 90292 .

• If driving, please park in the CYC visitor’s lot adjacent to Admiralty Way and accessed from the driveway which is between Café del Rey and the Warehouse Restaurant.
• Once parked, proceed by foot and enter the member’s parking lot by passing by the main entry gate.
• After passing the main entry gate, continue walking, bearing hard left along the chain link fence at the north end of the large parking lot until it ends at a dark green shade cloth covered rowing shed.
• We meet, at the rowing sheds which are opposite the tall bamboo plants separating the Warehouse Restaurant and CYC.

If you will be late or unable to attend, PLEASE text (and identify yourself) as soon as possible: Craig Leeds 310.948.1456 and/or Anna Wilczek 818.523.2987.

If the main entry gate is locked when you arrive, please text us.

Participants should bring: athletic shoes (running shoes), socks, shorts, shirts, sunscreen, water, a change of clothes,

towel and personal items (hats, sunglasses, orthotics, inhalers, etc).

To keep all the gear organized, a small duffel bag or back pack is recommended.

Clothes should fit reasonably tightly so while rowing, hands won’t get caught in shirts and shorts won’t get caught in the sliding seats.

Dressing in layers and using wicking type fabrics is recommended.

Here are some links that will be helpful (you may need to copy and paste into a browser):

• This is a video from Concept2 regarding technique on the rowing machine which is where we begin. Please watch the video before the first session.
• World Rowing
• This is a great pictorial representation of the rowing stroke as taught at CYC:
• This is a great video showing Australian world champion scullers and sweep rowers in a beautiful setting.
• Here’s a link to the US Women winning the quad at the 2015 World Championship:

Please don’t hesitate to ask for more information.

California Yacht Club-Junior Rowing

2021 J/70 WORLDS : CHAMPION

Peter duncan and relative obscurity return to the podium as j/70 world champion at cal yacht club.

2021 J/70 WORLDS : CORINTHIAN CHAMPIONS

The ducasse sailing team of santiago, chile, triumphed in the corinthian division..

2021 J/70 WORLDS : ONE PRO CHAMPIONS

Threatening minors wins one pro..

PETER DUNCAN AND RELATIVE OBSCURITY WIN THE 2021 J/70 WORLDS

Peter Duncan and Relative Obscurity Return to the Podium as J/70 World Champion at California Yacht Club

Peter Duncan’s Relative Obscurity has prevailed over 60 other teams and challenging wind conditions to capture the 2021 J/70 World Championship title at California Yacht Club, today. In a five-day series that tested the skill and patience of top-notch competitors from 11 nations, Duncan – sailing with Willem van Waay, Morgan Trubovich and Victor Diaz de Leon – secured a top five position in the beginning of the regatta and never let go.

“I’m elated!” said Duncan as he returned to the dock, bustling with activity. “That was a tough day out there. We didn’t start very well but had a bit of a break with a header on the first run of the second race that let us get close to everybody and sail through some folks we need to sail through,” he explained. “We have a lot of fun onboard – joke and laugh and keep it light – and that worked in our favor when we had to grind through. Everybody knows what their job is, and these guys do them exceptionally well.”

WELCOME TO THE J/70 WORLDS 2021

Cal Yacht Club is proud to host the 2021 racing of the J/70 Worlds.

RECAP / NEWS

Peter Duncan’s  Relative Obscurity  has prevailed over 60 other teams and challenging wind conditions to capture the 2021 J/70 World Championship title at California Yacht Club, today.

DAILY PHOTO / VIDEO

Threatening Minors  sailed by Jordan Janov, Grant Janov, Ryan Janov, Reddin Kherli and Willie Mcbride, took honors.

SOCIAL MEDIA

The J/70 is 22.75 feet with an 11 foot long cockpit and deck-stepped carbon mast for easy rigging and stepping.

The crew shall consist of 3 or more persons. The number of crew shall not be changed during an event.

The J/70 Class has been created as a strict one-design Class where the true test when racing is between crews and not boats and equipment.

Well, chances are it will not be like Day 1 of the Pre-Worlds. Maybe more like Day 2. But then again, this is MdR—anything can happen.

SANTA MONICA BAY

The Santa Monica Bay is dominated by an onshore breeze that typically doesn't kick in until around noon. Watch out for the oscillations!

CAL YACHT CLUB

CYC has a long history of excellence in race management. We host everything from Championship-level regattas to more casual weekend random leg races.

SAY WHAT? : THE J/70 WORLDS 2021

"Cal Yacht Club did an awesome job."

Peter Duncan | Relative Obscurity

“It was very cool that there were four boats who could have won this thing in the last race...”

CALIFORNIA YACHT CLUB

Founded originally in 1922 by yachtsmen including Charles Hathaway and Frank Garbutt, the California Yacht Club's first clubhouse was built in Wilmington, CA in Los Angeles' inner harbor (berths nos. 192 & 193), just opposite Terminal Island. Close by were the yards of renowned yacht builders Wilmington Boat Works and Fellows & Stewart (second location). In the club's inaugural year, member yachtsmen formed the first Star class fleet on the Pacific coast. Involved in all aspect of the sport, the club has encouraged a variety of pleasure boating, first in Los Angeles Harbor, and now in Santa Monica Bay.

Over the years CYC has been the club for numerous prominent yachtsmen including Merritt Adamson, Pierpont Davis, Roy E. Disney, J. Paul Getty, Samuel K. Rindge, William Stewart, James Kilroy and navigator extraodinaire, Ben Mitchell. Movie mogul Cecil B. DeMille once served as a trustee of the club and donated a gold cup for powerboat racing. Comedy film producers Al Christie and Hal Roach were both deeply involved in club activities in the twenties and thirties.

Power boating has always been a part of the club's mission along with sailing and rowing. The first CYC powerboat regatta was run in 1922 and the winner was none less than the famous Gar Wood in his Harmsworth Trophy winner Miss America. CYC's Catalina Challenge race for powerboats has been run annually since 1922.

A fire on Thanksgiving Day, 1930 severely damaged the original clubhouse although the heroic efforts of some members saved all of the trophies.

In 1932, several CYC members figured prominently in the sailing events at the Los Angeles Olympic Games. Owen Churchill, inventor of the SwimFin, won the gold medal in the Eight-Metre Class with Angelita.

Unfortunately, the club was forced to relinquish its key location in the East Basin to the Coast Guard for the war effort in 1941. A dormant period followed.

With the development of the long-awaited Marina Del Rey in the early 1960s, the club reformed in '63 and elected Fritz Overton, Commodore in 1923, as head of the "new" club. In 1966, they opened the modern clubhouse and marina facility that is their home today. The radial design of the building allows sweeping panoramic views of the marina.

The California Yacht Club is owned by the Hathaway family, owners of the Los Angeles Athletic Club. The annual "Great Catalina to Marina del Rey Rowing and Paddling Event" pays tribute to Charles Hathaway's row in 1976 from Catalina Island to the club on his 50th birthday.

The unique combination of private ownership and annually elected flag officers has worked well to establish California Yacht Club as one of the outstanding clubs in the nation.

‘Nothing left’: After California Yacht Club fire, residents mourn loss of a beloved spot

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In an instant, an overnight seaside blaze engulfed decades’ worth of boating trophies, historical artifacts and cherished memorabilia at the California Yacht Club in Marina del Rey on Monday.

Fire crews attempted to control the blaze as heavy smoke and flames consumed the two-story building. By the time they had subdued the fire two hours later, only the skeletal remnants of the clubhouse were left standing.

John Myers, senior vice president of the club, said the blaze had been reported by an employee working late in the clubhouse Monday night. The fire spared the remainder of the facilities on the ground, including the docks and the yachts moored there. But the clubhouse, and particularly its second floor, was all but wiped out.

“We are working closely with the Los Angeles County Fire Department in their investigation of the cause of the incident and will share those findings when they become available to us,” Myers said.

Members are left mourning, comparing the loss to the death of a loved one.

Jennifer Dakoske Koslu awoke in Rancho Mirage at 5:30 a.m. Tuesday, before the sun had risen, to find her phone inundated with text messages from club members.

The first message she read simply stated, “The CYC is gone.”

“As soon as I opened my phone, it went to a link on the Citizen app and saw a video of the club burning. I was shocked,” Dakoske Koslu said.

For the last 24 years, Dakoske Koslu and her family have been dedicated members of the CYC, whose clubhouse is a few miles away from their home in Playa del Rey. She said it is where her children have grown up, familiarizing themselves with every inch.

“I remember taking my son there on the Fourth of July when he was just 3 weeks old. It was the first place we went with him as a newborn,” Dakoske Koslu said.

She and her husband biked to the club in the aftermath of the fire, greeted by the charred remains of the building on Wednesday afternoon.

“The destruction is unbelievable. It’s clear that the fire was burning intensely on the second floor,” Dakoske Koslu said. “There’s nothing left.”

The second floor once housed a collection of the club’s prestigious racing trophies, kept on display for members and visitors. The fire melted all but a single salvageable California Cup. Most notably, the priceless King of Spain Trophy, acquired in 1929 from King Alfonso XIII, was lost.

Additionally, the club lost cherished photographs of every past commodore, a significant position within a yacht club. Members said they didn’t know if anyone had digitized the images of the commodores or of the club’s founders.

“We would tell yachting stories at the bar around lots of memorabilia, and the yachting artifacts behind the bar are all gone now,” Tom Materna said. “The yacht club provided us a facility for the off-the-water celebrations after hard-fought competition on the water.”

The CYC dates to the early 1920s, started by boat owners from the Los Angeles Athletic Club and other yacht clubs. The Board of Harbor Commissioners approved the first clubhouse in 1922, designed by famed architect Edwin Bergstrom, co-designer of the Pentagon.

In 1965, the yacht club submitted a proposal for an all-encompassing $1-million, two-story, 10,000-square-foot clubhouse on four acres off Admiralty Way. Members envisioned a state-of-the-art facility with 170 boat slips, a guest dock, a small boat hoist and a dry land storage facility for boats. The clubhouse that resulted was dedicated on June 10, 1967.

Then-Commodore William A. DeGroot Jr. told The Times that the triangular parcel of land on which the clubhouse still sits is a “perfectly logical place for a club facility, and a commanding view down the main channel of the marina.”

Though the building has historical significance to its members, it does not have a historic designation, according to Linda Dishman, president of the Los Angeles Conservancy.

“We are deeply saddened by this tragedy and so grateful for the outpouring of support from the community and our members,” Myers said. “CYC has been a beacon for the nautical community for the past 101 years.”

Materna, 68, first found out about the fire through Facebook as friends posted videos and photos of the damage Tuesday morning. Then he began receiving calls and text messages from friends.

“Everybody woke up in the morning and realized we’d lost a significant part of the sailing community,” Materna said.

His connection to the club dates back nearly 52 years, to when he was just 16 years old. After spending 30 years sailing professionally with Hobie Cats, mainly racing catamarans — a watercraft with two parallel hulls of equal size — he recently served as a crew member on other club members’ racing yachts.

The CYC is pivotal in the boat racing community, organizing and hosting events such as the Optimist National Championship and Junior Olympic trials, Materna said. He fondly remembers the hundreds of people from across the globe converging on the marina for similar events.

The main topic among members now is what’s next for the club. Dakoske Koslu noted that the club’s ownership changed over the last few years, and many are unsure and concerned about the club’s continuation after the fire.

The club relocated to the marina in 1967, leasing the land it sits on from the county.

“I don’t think the county has really valued the contributions of the California Yacht Club as an important part of the Marina. They value Trader Joe’s because it’s more money for them,” Dakoske Koslu said.

Dakoske Koslu said she’s seen numerous small marine-oriented businesses displaced from the marina, making way for more commercialized developments such as Trader Joe’s and Recreational Equipment Inc.

More to Read

Monterey Park shooting survivors dance again in sadness, solidarity: ‘Nothing can kill our spirit’

Jan. 22, 2024

Fire guts historic California Yacht Club in Marina del Rey

Dec. 12, 2023

After 24 days, officials declare Tustin hangar fire ‘fully extinguished’

Dec. 2, 2023

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Anthony De Leon is a 2023-24 reporting fellow at the Los Angeles Times. Born in Fresno to a Chicano family, he pursued his higher education in his hometown, earning an associate‘s degree in journalism from Fresno City College and then completing a bachelor’s in media, communications and journalism at Fresno State. He went on to complete his master’s in media innovation at the University of Nevada, Reno.

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Rowing club los angeles: everything you need to know.

If you’re interested in rowing and live in Los Angeles, you’re in luck. Los Angeles has a long history of rowing, and there are many rowing clubs to choose from. Rowing is an excellent way to stay in shape, meet new people, and enjoy the beautiful outdoors. In this article, we’ll take a closer look at rowing clubs in Los Angeles, their history, benefits, and how to join them.

History of Rowing Club in Los Angeles

Rowing has been a popular sport in Los Angeles for over a century, with the first rowing club, the Los Angeles Rowing Club, established in 1908. The club was founded by a group of rowing enthusiasts who wanted to promote the sport in Southern California. The first boathouse was located in the Venice Canals, and the club quickly gained popularity.

Over the years, many other rowing clubs have been established in Los Angeles, including the California Yacht Club, the Long Beach Rowing Association, and the Marina Aquatic Center. These clubs have played a significant role in promoting rowing in Los Angeles, and their members have achieved many successes in local and national competitions. Despite many challenges, including funding, maintenance, and environmental issues, rowing clubs in Los Angeles continue to thrive and attract new members.

Rowing is a low-impact, full-body workout that is suitable for people of all ages and fitness levels. Joining a rowing club in Los Angeles offers many benefits, including improving your physical and mental health, socializing with like-minded people, and competing in rowing events.

Health Benefits

Rowing is an excellent way to improve your cardiovascular health, build muscle strength, and burn calories. According to a study by Harvard Health Publishing, rowing burns more calories than running or cycling, making it an effective way to lose weight and get in shape. Rowing also helps to reduce stress, improve sleep quality, and boost your mood.

Social Benefits

Joining a rowing club in Los Angeles provides an opportunity to meet new people, make friends, and socialize. Rowing clubs often organize social events such as barbecues, parties, and team-building exercises, allowing you to connect with other members outside of the boat. Rowing also promotes teamwork, communication, and leadership skills, which can be applied to other areas of your life.

Next, we will discuss how to join a rowing club in LA and popular rowing clubs in the area.

The Benefits of Joining a Rowing Club in Los Angeles

Rowing is an excellent way to stay active and healthy, but joining a rowing club in Los Angeles offers many additional benefits beyond physical fitness.

Rowing is a low-impact, full-body workout that improves cardiovascular health, builds muscle strength, and burns calories. Regular rowing can help reduce the risk of chronic diseases such as diabetes, heart disease, and obesity. Additionally, rowing is an excellent way to improve flexibility and balance, which is essential for maintaining mobility as you age.

Joining a rowing club in Los Angeles provides an opportunity to meet new people, make friends, and develop a sense of community. Rowing clubs often organize social events such as barbecues, parties, and team-building exercises, allowing you to connect with other members outside of the boat. Rowing also promotes teamwork, communication, and leadership skills, which can be applied to other areas of your life.

Competitive Benefits

For those who enjoy competition, joining a rowing club in Los Angeles provides opportunities to participate in local and national rowing events. Rowing clubs often have teams that compete in regattas, races, and other events, giving members a chance to test their skills against other rowers. Participating in rowing competitions can be a fun and rewarding experience, and it can help you set and achieve personal goals.

How to Join a Rowing Club in Los Angeles

Joining a rowing club in Los Angeles is relatively easy, but there are some requirements you need to meet before you can become a member. Here’s what you need to know:

Requirements for Joining

Most rowing clubs in Los Angeles require that you be at least 18 years old and have a basic level of fitness. Some clubs may also require that you have previous rowing experience or complete a rowing class before you can join. Additionally, you may need to pass a swim test to ensure that you can safely navigate the water.

Application Process

To join a rowing club in Los Angeles, you will need to fill out an application form and pay a membership fee. The application form will typically ask for your contact information, rowing experience (if any), and emergency contact information. Once your application is approved, you will be given access to the club’s facilities and equipment. Some clubs may also require that you attend an orientation session to learn about the club’s rules and safety procedures.

Popular Rowing Clubs in LA

There are many rowing clubs to choose from in Los Angeles, each with its unique features, facilities, and programs. Here are some of the most popular rowing clubs in the area:

1. Los Angeles Rowing Club

The Los Angeles Rowing Club is the oldest rowing club in the city, founded in 1908. The club is located in a historic boathouse in the Marina del Rey harbor and offers programs for rowers of all levels, from beginners to advanced. The club has a competitive racing team, a recreational rowing program, and hosts many social events throughout the year.

2. California Yacht Club

The California Yacht Club is located in Marina del Rey and has a dedicated rowing program with a focus on competitive racing. The club has a state-of-the-art boathouse and equipment and offers programs for juniors and adults.

3. Long Beach Rowing Association

The Long Beach Rowing Association is located in Long Beach and offers programs for all levels, including recreational, competitive, and adaptive rowing. The club has a large boathouse and a fleet of boats, including singles, doubles, quads, and eights.

4. Marina Aquatic Center

The Marina Aquatic Center is located in Marina del Rey and is affiliated with the University of Southern California. The center offers rowing programs for USC students and the general public, including recreational and competitive rowing. The center has a large boathouse, a fleet of boats, and a team of experienced coaches.

In conclusion, joining a rowing club in Los Angeles is an excellent way to stay in shape, make new friends, and enjoy the beautiful outdoors. Rowing offers many health benefits, including improving cardiovascular health, building muscle strength, and reducing stress. By joining a rowing club, you can also develop teamwork, communication, and leadership skills, which can be applied to other areas of your life.

If you’re interested in joining a rowing club in Los Angeles, there are many options to choose from, each with its unique features and programs. We encourage you to check out the clubs we’ve listed and find one that suits your needs and interests. Remember, rowing is a lifelong sport that can be enjoyed at any age and fitness level, so why not give it a try and see what it can do for you? Join a rowing club in Los Angeles today and experience the benefits of this fantastic sport!

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"They fell to the ground with screams": Russian Guards fired at children single near Moscow - there is a casualty

2023-08-20T20:58:57.477Z

Highlights: In Russia, in the city of Elektrostal (Moscow region), during demonstrations, Rosgvardia soldiers began shooting at spectators with children from machine guns with blank cartridges. One child received serious damage from a rebounded cartridge case. In the video, a child can be heard crying and screaming violently. It is also interesting that Russia recently arranged a solemn farewell to Vladimir Shestakov, convicted for the murder of a child, who became a mercenary of PMC "Wagner" and was liquidated in the war in Ukraine.

In Russia, in the city of Elektrostal (Moscow region), during demonstrations, Rosgvardia soldiers began shooting at spectators with children from machine guns with blank cartridges.

So far, one injured child is known.

This was reported by the local Telegram channel of the Cheka-OGPU.

"Small children were clutching their heads screaming and falling to the ground. Not without injuries. The child received serious damage from a rebounded cartridge case," the report said.

One of the witnesses to the incident posted a video. It was her child who was shot by the Russian Guards. In the video, a child can be heard crying and screaming violently.

After the woman realized that her child had been wounded, she called her husband and doctor.

Meanwhile, Russian occupier Ivan Alekseev in the war in Ukraine after a drunken quarrel killed his colleague and tried to cover up the crime, saying it was the work of "Ukrainian saboteurs."

It is also interesting that Russia recently arranged a solemn farewell to Vladimir Shestakov, convicted for the murder of a child, who became a mercenary of PMC "Wagner" and was liquidated in the war in Ukraine.

• The suspect in the murder of a military volunteer was released from custody
• They will teach "patriotism": Russians in the occupied territories launch cadet classes
• Russia has created another training ground near Mariupol: how many soldiers are in the city

Source: tsn

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Field hockey - Dinamo Elektrostal Moscow

Hockey Club Dinamo Elektrostal is a field hockey team from Russia, based in Moscow. The club was founded in 1994.

Dinamo Elektrostal Moscow - Results

2021/2022 2018/2019 2017/2018 2017 2015/2016 2013/2014 2011/2012 2007/2008

Men's Euro Hockey League - Final Round - 2021/2022

Dinamo elektrostal moscow - identity.

• Official name : Hockey Club Dinamo Elektrostal
• Country : Russia
• Location : Moscow
• Founded : 1994
• Wikipedia link : http://nl.wikipedia.org/wiki/Dinamo_Elektrostal

Dinamo Elektrostal Moscow - Titles, trophies and places of honor

• Best result : First Round in 2021/2022
• Best result : 1st
• 1 times first in 2010
• 1 times second in 2009
• 1 times third in 2017

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Club Risovalshchikov

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CLUB RISOVALSHCHIKOV, Moscow - Butovo South - Restaurant Reviews, Photos & Phone Number - Tripadvisor

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Yacht Rock: ELLE’s February 2024 Shopping Guide

From the rowing club to the runway, preppy classics are making a splash.

It’s time to recruit your own team of sporty staples. Go all in on the throwback aesthetic by pairing collegiate standbys with rope bracelets and loafers.

Dries Van Noten Retro Runner Sneakers

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California Yacht Club was established in 1922 and boated its' first competitive rowing team back in the 1930's. In 1977, after a long hiatus, Stan Mullin, Ken Jacobs and Charles Hathaway reactivated rowing at the Club. CYC rowers now number about 60, and represent all abilities, ages and motivations.

Adult Rowing - California Yacht Club CYC is a great place for adults of all ages and skills to row out of Marina Del Rey. We have more than 60 active adult rowers with a wide range of interests and motivations varying from recreational, fitness, social, open water, and racing.

CYC Rowing - California Yacht Club - Rowing About us CYC Junior Rowing CYC offers a range of rowing programs suitable for girls and boys ages 12 - 18. CYC membership is not required to participate. Our junior rowing programs include: a highly successful competitive rowing team, a recreational program and private lessons.

Everyone at the California Yacht Club is devastated by the fire that took place at the structure of our historic club on Monday, December 11, 2023. We want to extend our sincere thanks to the Los Angeles County Fire Department for their rapid response and intensive efforts to extinguish the fire. We are committed to working with the fire ...

Junior Sailing and Rowing Program. Open to all youths in the community (ages 8-18). Full time Junior staff. Comments: Active power, sail and rowing fleets. Family friendly Club with a full schedule of races, cruises, and social activities.

California Yacht Club's Junior Rowing Program runs seasonally from September through June, and extends into Summer with several two week "Learn to Row" camps. The core instruction of the program is sculling. Singles, doubles and quads are the on the water tools for quality learning, and our inventory of ergometers are used for land training.

California Yacht Club, Marina del Rey, California. 4,778 likes · 133 talking about this · 24,313 were here. The California Yacht Club mission is to...

California Yacht Club was established in 1922 and boated its first competitive rowing team back in the 1930's. In 1977, after a long hiatus, Stan Mullin, Ken Jacobs and Charles Hathaway reactivated rowing at the Club and it has grown since then.

The California Yacht Club has a wonderful community of members who enjoy a packed social calendar of over 400 events throughout the year that features water activities, wine tastings, movies...

California Yacht Club is located at: 4469 Admiralty Way, Marina del Rey, CA 90292. ... California Yacht Club-Junior Rowing. Craig Leeds [email protected] 310.948.1456. and. Anna Wilczek [email protected] 818.523.2987. CYC offers a variety of summer programs for junior age (13 - 18) boys and girls. CYC membership is not required to enroll.

J/70 Worlds 2021 : Peter Duncan's Relative Obscurity has prevailed over 60 other teams and challenging wind conditions to capture the 2021 J/70 World Championship title at California Yacht Club, today. In a five-day series that tested the skill and patience of top-notch competitors from 11 nations, Duncan - sailing with Willem van Waay, Morgan Trubovich and Victor Diaz de Leon - secured ...

Congratulations to the California Yacht Club Rowing Team on an impressive showing at the Long Beach Christmas regatta! 27 medals is an incredible achievement, and we are so proud of all of your hard...

The California Yacht Club is owned by the Hathaway family, owners of the Los Angeles Athletic Club. The annual "Great Catalina to Marina del Rey Rowing and Paddling Event" pays tribute to Charles Hathaway's row in 1976 from Catalina Island to the club on his 50th birthday.

3,429 Followers, 1,052 Following, 637 Posts - See Instagram photos and videos from California Yacht Club (@calyachtclub) 3,429 Followers, 1,052 Following, 637 Posts - See Instagram photos and videos from California Yacht Club (@calyachtclub) Something went wrong. There's an issue and the page could not be loaded. ...

515 Followers, 265 Following, 67 Posts - See Instagram photos and videos from California Yacht Club Rowing (@calyachtclubrowing)

• 1st US Rowing Coach to win back-to-back Gold medals in the Mens Eight (97,98,99) (04,05). ... RC and California Rowing Club • Advisory coach at Cambridge University in England for their annual Boat Race since 1993 • U.S. Men's Olympic Head Coach from 2009-2012 London Olympic Games (Bronze - M4-)

When the California Yacht Club burned in Marina del Rey, prestigious awards, a priceless trophy and cherished photos were lost. Members mourn the club's loss.

The Los Angeles Rowing Club is the oldest rowing club in the city, founded in 1908. The club is located in a historic boathouse in the Marina del Rey harbor and offers programs for rowers of all levels, from beginners to advanced. The club has a competitive racing team, a recreational rowing program, and hosts many social events throughout the ...

Related Articles J/Teams Silver in Southern California Islands Race J/130 1st, J/111 2nd, & J/125 2nd The annual Southern California offshore and coastal racing season kicked off with the 2024 Islands Race, a coastal race co-hosted by Newport Harbor Yacht Club and San Diego Yacht Club since 2010. Posted on 19 Feb BadPak wins 2024 Islands Race Rio100 sets course record The annual Southern ...

The moment of the explosion in Moscow was broadcast on the rowing championship: video. News/Politics 2023-08-11T12:06:55.387Z. 8-Year-Old Boy Killed in Russian Strike on Western Ukraine. News/Politics 2023-08-11T14:26:57.994Z. They were going to reconnaissance the territory: border guards "landed" the occupiers' UAVs near Avdiivka.

Dinamo Elektrostal Moscow - Titles, trophies and places of honor. Men's Euro Hockey League since 2007/2008 (7 participations) . Best result : First Round in 2021/2022; EuroHockey Men's Club Trophy since 2008 . Best result : 1st

Club Risovalshchikov, Moscow: See 7 unbiased reviews of Club Risovalshchikov, rated 4.5 of 5 on Tripadvisor and ranked #4,585 of 15,890 restaurants in Moscow.

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how to determine the hypothesis in statistics

Statistics Made Easy

Introduction to Hypothesis Testing

A statistical hypothesis is an assumption about a population parameter .

For example, we may assume that the mean height of a male in the U.S. is 70 inches.

The assumption about the height is the statistical hypothesis and the true mean height of a male in the U.S. is the population parameter .

A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical hypothesis.

The Two Types of Statistical Hypotheses

To test whether a statistical hypothesis about a population parameter is true, we obtain a random sample from the population and perform a hypothesis test on the sample data.

There are two types of statistical hypotheses:

The null hypothesis , denoted as H 0 , is the hypothesis that the sample data occurs purely from chance.

The alternative hypothesis , denoted as H 1 or H a , is the hypothesis that the sample data is influenced by some non-random cause.

Hypothesis Tests

A hypothesis test consists of five steps:

1. State the hypotheses. 

State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false.

2. Determine a significance level to use for the hypothesis.

Decide on a significance level. Common choices are .01, .05, and .1. 

3. Find the test statistic.

Find the test statistic and the corresponding p-value. Often we are analyzing a population mean or proportion and the general formula to find the test statistic is: (sample statistic – population parameter) / (standard deviation of statistic)

4. Reject or fail to reject the null hypothesis.

Using the test statistic or the p-value, determine if you can reject or fail to reject the null hypothesis based on the significance level.

The p-value  tells us the strength of evidence in support of a null hypothesis. If the p-value is less than the significance level, we reject the null hypothesis.

5. Interpret the results. 

Interpret the results of the hypothesis test in the context of the question being asked. 

The Two Types of Decision Errors

There are two types of decision errors that one can make when doing a hypothesis test:

Type I error: You reject the null hypothesis when it is actually true. The probability of committing a Type I error is equal to the significance level, often called  alpha , and denoted as α.

Type II error: You fail to reject the null hypothesis when it is actually false. The probability of committing a Type II error is called the Power of the test or  Beta , denoted as β.

One-Tailed and Two-Tailed Tests

A statistical hypothesis can be one-tailed or two-tailed.

A one-tailed hypothesis involves making a “greater than” or “less than ” statement.

For example, suppose we assume the mean height of a male in the U.S. is greater than or equal to 70 inches. The null hypothesis would be H0: µ ≥ 70 inches and the alternative hypothesis would be Ha: µ < 70 inches.

A two-tailed hypothesis involves making an “equal to” or “not equal to” statement.

For example, suppose we assume the mean height of a male in the U.S. is equal to 70 inches. The null hypothesis would be H0: µ = 70 inches and the alternative hypothesis would be Ha: µ ≠ 70 inches.

Note: The “equal” sign is always included in the null hypothesis, whether it is =, ≥, or ≤.

Related:   What is a Directional Hypothesis?

Types of Hypothesis Tests

There are many different types of hypothesis tests you can perform depending on the type of data you’re working with and the goal of your analysis.

The following tutorials provide an explanation of the most common types of hypothesis tests:

Introduction to the One Sample t-test Introduction to the Two Sample t-test Introduction to the Paired Samples t-test Introduction to the One Proportion Z-Test Introduction to the Two Proportion Z-Test

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S.3 hypothesis testing.

In reviewing hypothesis tests, we start first with the general idea. Then, we keep returning to the basic procedures of hypothesis testing, each time adding a little more detail.

The general idea of hypothesis testing involves:

• Making an initial assumption.
• Collecting evidence (data).
• Based on the available evidence (data), deciding whether to reject or not reject the initial assumption.

Every hypothesis test — regardless of the population parameter involved — requires the above three steps.

Example S.3.1

Is normal body temperature really 98.6 degrees f section  .

Consider the population of many, many adults. A researcher hypothesized that the average adult body temperature is lower than the often-advertised 98.6 degrees F. That is, the researcher wants an answer to the question: "Is the average adult body temperature 98.6 degrees? Or is it lower?" To answer his research question, the researcher starts by assuming that the average adult body temperature was 98.6 degrees F.

Then, the researcher went out and tried to find evidence that refutes his initial assumption. In doing so, he selects a random sample of 130 adults. The average body temperature of the 130 sampled adults is 98.25 degrees.

Then, the researcher uses the data he collected to make a decision about his initial assumption. It is either likely or unlikely that the researcher would collect the evidence he did given his initial assumption that the average adult body temperature is 98.6 degrees:

• If it is likely , then the researcher does not reject his initial assumption that the average adult body temperature is 98.6 degrees. There is not enough evidence to do otherwise.
• either the researcher's initial assumption is correct and he experienced a very unusual event;
• or the researcher's initial assumption is incorrect.

In statistics, we generally don't make claims that require us to believe that a very unusual event happened. That is, in the practice of statistics, if the evidence (data) we collected is unlikely in light of the initial assumption, then we reject our initial assumption.

Example S.3.2

Criminal trial analogy section  .

One place where you can consistently see the general idea of hypothesis testing in action is in criminal trials held in the United States. Our criminal justice system assumes "the defendant is innocent until proven guilty." That is, our initial assumption is that the defendant is innocent.

In the practice of statistics, we make our initial assumption when we state our two competing hypotheses -- the null hypothesis ( H 0 ) and the alternative hypothesis ( H A ). Here, our hypotheses are:

• H 0 : Defendant is not guilty (innocent)
• H A : Defendant is guilty

In statistics, we always assume the null hypothesis is true . That is, the null hypothesis is always our initial assumption.

The prosecution team then collects evidence — such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, and handwriting samples — with the hopes of finding "sufficient evidence" to make the assumption of innocence refutable.

In statistics, the data are the evidence.

The jury then makes a decision based on the available evidence:

• If the jury finds sufficient evidence — beyond a reasonable doubt — to make the assumption of innocence refutable, the jury rejects the null hypothesis and deems the defendant guilty. We behave as if the defendant is guilty.
• If there is insufficient evidence, then the jury does not reject the null hypothesis . We behave as if the defendant is innocent.

In statistics, we always make one of two decisions. We either "reject the null hypothesis" or we "fail to reject the null hypothesis."

Errors in Hypothesis Testing Section  

Did you notice the use of the phrase "behave as if" in the previous discussion? We "behave as if" the defendant is guilty; we do not "prove" that the defendant is guilty. And, we "behave as if" the defendant is innocent; we do not "prove" that the defendant is innocent.

This is a very important distinction! We make our decision based on evidence not on 100% guaranteed proof. Again:

• If we reject the null hypothesis, we do not prove that the alternative hypothesis is true.
• If we do not reject the null hypothesis, we do not prove that the null hypothesis is true.

We merely state that there is enough evidence to behave one way or the other. This is always true in statistics! Because of this, whatever the decision, there is always a chance that we made an error .

Let's review the two types of errors that can be made in criminal trials:

Table S.3.2 shows how this corresponds to the two types of errors in hypothesis testing.

Note that, in statistics, we call the two types of errors by two different  names -- one is called a "Type I error," and the other is called  a "Type II error." Here are the formal definitions of the two types of errors:

There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!

Making the Decision Section  

Recall that it is either likely or unlikely that we would observe the evidence we did given our initial assumption. If it is likely , we do not reject the null hypothesis. If it is unlikely , then we reject the null hypothesis in favor of the alternative hypothesis. Effectively, then, making the decision reduces to determining "likely" or "unlikely."

In statistics, there are two ways to determine whether the evidence is likely or unlikely given the initial assumption:

• We could take the " critical value approach " (favored in many of the older textbooks).
• Or, we could take the " P -value approach " (what is used most often in research, journal articles, and statistical software).

In the next two sections, we review the procedures behind each of these two approaches. To make our review concrete, let's imagine that μ is the average grade point average of all American students who major in mathematics. We first review the critical value approach for conducting each of the following three hypothesis tests about the population mean $\mu$:

In Practice

• We would want to conduct the first hypothesis test if we were interested in concluding that the average grade point average of the group is more than 3.
• We would want to conduct the second hypothesis test if we were interested in concluding that the average grade point average of the group is less than 3.
• And, we would want to conduct the third hypothesis test if we were only interested in concluding that the average grade point average of the group differs from 3 (without caring whether it is more or less than 3).

Upon completing the review of the critical value approach, we review the P -value approach for conducting each of the above three hypothesis tests about the population mean \(\mu\). The procedures that we review here for both approaches easily extend to hypothesis tests about any other population parameter.

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Everything you need to know about the normal distribution, an in-depth explanation of cumulative distribution function, a complete guide to chi-square test, a complete guide on hypothesis testing in statistics, understanding the fundamentals of arithmetic and geometric progression, the definitive guide to understand spearman’s rank correlation, a comprehensive guide to understand mean squared error, all you need to know about the empirical rule in statistics, the complete guide to skewness and kurtosis, a holistic look at bernoulli distribution.

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In today’s data-driven world , decisions are based on data all the time. Hypothesis plays a crucial role in that process, whether it may be making business decisions, in the health sector, academia, or in quality improvement. Without hypothesis & hypothesis tests, you risk drawing the wrong conclusions and making bad decisions. In this tutorial, you will look at Hypothesis Testing in Statistics.

What Is Hypothesis Testing in Statistics?

Hypothesis Testing is a type of statistical analysis in which you put your assumptions about a population parameter to the test. It is used to estimate the relationship between 2 statistical variables.

Let's discuss few examples of statistical hypothesis from real-life - 

• A teacher assumes that 60% of his college's students come from lower-middle-class families.
• A doctor believes that 3D (Diet, Dose, and Discipline) is 90% effective for diabetic patients.

Now that you know about hypothesis testing, look at the two types of hypothesis testing in statistics.

Hypothesis Testing Formula

Z = ( x̅ – μ0 ) / (σ /√n)

• Here, x̅ is the sample mean,
• μ0 is the population mean,
• σ is the standard deviation,
• n is the sample size.

How Hypothesis Testing Works?

An analyst performs hypothesis testing on a statistical sample to present evidence of the plausibility of the null hypothesis. Measurements and analyses are conducted on a random sample of the population to test a theory. Analysts use a random population sample to test two hypotheses: the null and alternative hypotheses.

The null hypothesis is typically an equality hypothesis between population parameters; for example, a null hypothesis may claim that the population means return equals zero. The alternate hypothesis is essentially the inverse of the null hypothesis (e.g., the population means the return is not equal to zero). As a result, they are mutually exclusive, and only one can be correct. One of the two possibilities, however, will always be correct.

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Null Hypothesis and Alternate Hypothesis

The Null Hypothesis is the assumption that the event will not occur. A null hypothesis has no bearing on the study's outcome unless it is rejected.

H0 is the symbol for it, and it is pronounced H-naught.

The Alternate Hypothesis is the logical opposite of the null hypothesis. The acceptance of the alternative hypothesis follows the rejection of the null hypothesis. H1 is the symbol for it.

Let's understand this with an example.

A sanitizer manufacturer claims that its product kills 95 percent of germs on average. 

To put this company's claim to the test, create a null and alternate hypothesis.

H0 (Null Hypothesis): Average = 95%.

Alternative Hypothesis (H1): The average is less than 95%.

Another straightforward example to understand this concept is determining whether or not a coin is fair and balanced. The null hypothesis states that the probability of a show of heads is equal to the likelihood of a show of tails. In contrast, the alternate theory states that the probability of a show of heads and tails would be very different.

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Hypothesis Testing Calculation With Examples

Let's consider a hypothesis test for the average height of women in the United States. Suppose our null hypothesis is that the average height is 5'4". We gather a sample of 100 women and determine that their average height is 5'5". The standard deviation of population is 2.

To calculate the z-score, we would use the following formula:

z = ( x̅ – μ0 ) / (σ /√n)

z = (5'5" - 5'4") / (2" / √100)

z = 0.5 / (0.045)

 We will reject the null hypothesis as the z-score of 11.11 is very large and conclude that there is evidence to suggest that the average height of women in the US is greater than 5'4".

Steps of Hypothesis Testing

Step 1: specify your null and alternate hypotheses.

It is critical to rephrase your original research hypothesis (the prediction that you wish to study) as a null (Ho) and alternative (Ha) hypothesis so that you can test it quantitatively. Your first hypothesis, which predicts a link between variables, is generally your alternate hypothesis. The null hypothesis predicts no link between the variables of interest.

Step 2: Gather Data

For a statistical test to be legitimate, sampling and data collection must be done in a way that is meant to test your hypothesis. You cannot draw statistical conclusions about the population you are interested in if your data is not representative.

Step 3: Conduct a Statistical Test

Other statistical tests are available, but they all compare within-group variance (how to spread out the data inside a category) against between-group variance (how different the categories are from one another). If the between-group variation is big enough that there is little or no overlap between groups, your statistical test will display a low p-value to represent this. This suggests that the disparities between these groups are unlikely to have occurred by accident. Alternatively, if there is a large within-group variance and a low between-group variance, your statistical test will show a high p-value. Any difference you find across groups is most likely attributable to chance. The variety of variables and the level of measurement of your obtained data will influence your statistical test selection.

Step 4: Determine Rejection Of Your Null Hypothesis

Your statistical test results must determine whether your null hypothesis should be rejected or not. In most circumstances, you will base your judgment on the p-value provided by the statistical test. In most circumstances, your preset level of significance for rejecting the null hypothesis will be 0.05 - that is, when there is less than a 5% likelihood that these data would be seen if the null hypothesis were true. In other circumstances, researchers use a lower level of significance, such as 0.01 (1%). This reduces the possibility of wrongly rejecting the null hypothesis.

Step 5: Present Your Results 

The findings of hypothesis testing will be discussed in the results and discussion portions of your research paper, dissertation, or thesis. You should include a concise overview of the data and a summary of the findings of your statistical test in the results section. You can talk about whether your results confirmed your initial hypothesis or not in the conversation. Rejecting or failing to reject the null hypothesis is a formal term used in hypothesis testing. This is likely a must for your statistics assignments.

Types of Hypothesis Testing

To determine whether a discovery or relationship is statistically significant, hypothesis testing uses a z-test. It usually checks to see if two means are the same (the null hypothesis). Only when the population standard deviation is known and the sample size is 30 data points or more, can a z-test be applied.

A statistical test called a t-test is employed to compare the means of two groups. To determine whether two groups differ or if a procedure or treatment affects the population of interest, it is frequently used in hypothesis testing.

Chi-Square 

You utilize a Chi-square test for hypothesis testing concerning whether your data is as predicted. To determine if the expected and observed results are well-fitted, the Chi-square test analyzes the differences between categorical variables from a random sample. The test's fundamental premise is that the observed values in your data should be compared to the predicted values that would be present if the null hypothesis were true.

Hypothesis Testing and Confidence Intervals

Both confidence intervals and hypothesis tests are inferential techniques that depend on approximating the sample distribution. Data from a sample is used to estimate a population parameter using confidence intervals. Data from a sample is used in hypothesis testing to examine a given hypothesis. We must have a postulated parameter to conduct hypothesis testing.

Bootstrap distributions and randomization distributions are created using comparable simulation techniques. The observed sample statistic is the focal point of a bootstrap distribution, whereas the null hypothesis value is the focal point of a randomization distribution.

A variety of feasible population parameter estimates are included in confidence ranges. In this lesson, we created just two-tailed confidence intervals. There is a direct connection between these two-tail confidence intervals and these two-tail hypothesis tests. The results of a two-tailed hypothesis test and two-tailed confidence intervals typically provide the same results. In other words, a hypothesis test at the 0.05 level will virtually always fail to reject the null hypothesis if the 95% confidence interval contains the predicted value. A hypothesis test at the 0.05 level will nearly certainly reject the null hypothesis if the 95% confidence interval does not include the hypothesized parameter.

Simple and Composite Hypothesis Testing

Depending on the population distribution, you can classify the statistical hypothesis into two types.

Simple Hypothesis: A simple hypothesis specifies an exact value for the parameter.

Composite Hypothesis: A composite hypothesis specifies a range of values.

A company is claiming that their average sales for this quarter are 1000 units. This is an example of a simple hypothesis.

Suppose the company claims that the sales are in the range of 900 to 1000 units. Then this is a case of a composite hypothesis.

One-Tailed and Two-Tailed Hypothesis Testing

The One-Tailed test, also called a directional test, considers a critical region of data that would result in the null hypothesis being rejected if the test sample falls into it, inevitably meaning the acceptance of the alternate hypothesis.

In a one-tailed test, the critical distribution area is one-sided, meaning the test sample is either greater or lesser than a specific value.

In two tails, the test sample is checked to be greater or less than a range of values in a Two-Tailed test, implying that the critical distribution area is two-sided.

If the sample falls within this range, the alternate hypothesis will be accepted, and the null hypothesis will be rejected.

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Right Tailed Hypothesis Testing

If the larger than (>) sign appears in your hypothesis statement, you are using a right-tailed test, also known as an upper test. Or, to put it another way, the disparity is to the right. For instance, you can contrast the battery life before and after a change in production. Your hypothesis statements can be the following if you want to know if the battery life is longer than the original (let's say 90 hours):

• The null hypothesis is (H0 <= 90) or less change.
• A possibility is that battery life has risen (H1) > 90.

The crucial point in this situation is that the alternate hypothesis (H1), not the null hypothesis, decides whether you get a right-tailed test.

Left Tailed Hypothesis Testing

Alternative hypotheses that assert the true value of a parameter is lower than the null hypothesis are tested with a left-tailed test; they are indicated by the asterisk "<".

Suppose H0: mean = 50 and H1: mean not equal to 50

According to the H1, the mean can be greater than or less than 50. This is an example of a Two-tailed test.

In a similar manner, if H0: mean >=50, then H1: mean <50

Here the mean is less than 50. It is called a One-tailed test.

Type 1 and Type 2 Error

A hypothesis test can result in two types of errors.

Type 1 Error: A Type-I error occurs when sample results reject the null hypothesis despite being true.

Type 2 Error: A Type-II error occurs when the null hypothesis is not rejected when it is false, unlike a Type-I error.

Suppose a teacher evaluates the examination paper to decide whether a student passes or fails.

H0: Student has passed

H1: Student has failed

Type I error will be the teacher failing the student [rejects H0] although the student scored the passing marks [H0 was true]. 

Type II error will be the case where the teacher passes the student [do not reject H0] although the student did not score the passing marks [H1 is true].

Level of Significance

The alpha value is a criterion for determining whether a test statistic is statistically significant. In a statistical test, Alpha represents an acceptable probability of a Type I error. Because alpha is a probability, it can be anywhere between 0 and 1. In practice, the most commonly used alpha values are 0.01, 0.05, and 0.1, which represent a 1%, 5%, and 10% chance of a Type I error, respectively (i.e. rejecting the null hypothesis when it is in fact correct).

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A p-value is a metric that expresses the likelihood that an observed difference could have occurred by chance. As the p-value decreases the statistical significance of the observed difference increases. If the p-value is too low, you reject the null hypothesis.

Here you have taken an example in which you are trying to test whether the new advertising campaign has increased the product's sales. The p-value is the likelihood that the null hypothesis, which states that there is no change in the sales due to the new advertising campaign, is true. If the p-value is .30, then there is a 30% chance that there is no increase or decrease in the product's sales.  If the p-value is 0.03, then there is a 3% probability that there is no increase or decrease in the sales value due to the new advertising campaign. As you can see, the lower the p-value, the chances of the alternate hypothesis being true increases, which means that the new advertising campaign causes an increase or decrease in sales.

Why is Hypothesis Testing Important in Research Methodology?

Hypothesis testing is crucial in research methodology for several reasons:

• Provides evidence-based conclusions: It allows researchers to make objective conclusions based on empirical data, providing evidence to support or refute their research hypotheses.
• Supports decision-making: It helps make informed decisions, such as accepting or rejecting a new treatment, implementing policy changes, or adopting new practices.
• Adds rigor and validity: It adds scientific rigor to research using statistical methods to analyze data, ensuring that conclusions are based on sound statistical evidence.
• Contributes to the advancement of knowledge: By testing hypotheses, researchers contribute to the growth of knowledge in their respective fields by confirming existing theories or discovering new patterns and relationships.

Limitations of Hypothesis Testing

Hypothesis testing has some limitations that researchers should be aware of:

• It cannot prove or establish the truth: Hypothesis testing provides evidence to support or reject a hypothesis, but it cannot confirm the absolute truth of the research question.
• Results are sample-specific: Hypothesis testing is based on analyzing a sample from a population, and the conclusions drawn are specific to that particular sample.
• Possible errors: During hypothesis testing, there is a chance of committing type I error (rejecting a true null hypothesis) or type II error (failing to reject a false null hypothesis).
• Assumptions and requirements: Different tests have specific assumptions and requirements that must be met to accurately interpret results.

After reading this tutorial, you would have a much better understanding of hypothesis testing, one of the most important concepts in the field of Data Science . The majority of hypotheses are based on speculation about observed behavior, natural phenomena, or established theories.

If you are interested in statistics of data science and skills needed for such a career, you ought to explore Simplilearn’s Post Graduate Program in Data Science.

If you have any questions regarding this ‘Hypothesis Testing In Statistics’ tutorial, do share them in the comment section. Our subject matter expert will respond to your queries. Happy learning!

1. What is hypothesis testing in statistics with example?

Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence. An example: testing if a new drug improves patient recovery (Ha) compared to the standard treatment (H0) based on collected patient data.

2. What is hypothesis testing and its types?

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. It involves formulating two hypotheses: the null hypothesis (H0), which represents the default assumption, and the alternative hypothesis (Ha), which contradicts H0. The goal is to assess the evidence and determine whether there is enough statistical significance to reject the null hypothesis in favor of the alternative hypothesis.

Types of hypothesis testing:

• One-sample test: Used to compare a sample to a known value or a hypothesized value.
• Two-sample test: Compares two independent samples to assess if there is a significant difference between their means or distributions.
• Paired-sample test: Compares two related samples, such as pre-test and post-test data, to evaluate changes within the same subjects over time or under different conditions.
• Chi-square test: Used to analyze categorical data and determine if there is a significant association between variables.
• ANOVA (Analysis of Variance): Compares means across multiple groups to check if there is a significant difference between them.

3. What are the steps of hypothesis testing?

The steps of hypothesis testing are as follows:

• Formulate the hypotheses: State the null hypothesis (H0) and the alternative hypothesis (Ha) based on the research question.
• Set the significance level: Determine the acceptable level of error (alpha) for making a decision.
• Collect and analyze data: Gather and process the sample data.
• Compute test statistic: Calculate the appropriate statistical test to assess the evidence.
• Make a decision: Compare the test statistic with critical values or p-values and determine whether to reject H0 in favor of Ha or not.
• Draw conclusions: Interpret the results and communicate the findings in the context of the research question.

4. What are the 2 types of hypothesis testing?

• One-tailed (or one-sided) test: Tests for the significance of an effect in only one direction, either positive or negative.
• Two-tailed (or two-sided) test: Tests for the significance of an effect in both directions, allowing for the possibility of a positive or negative effect.

The choice between one-tailed and two-tailed tests depends on the specific research question and the directionality of the expected effect.

5. What are the 3 major types of hypothesis?

The three major types of hypotheses are:

• Null Hypothesis (H0): Represents the default assumption, stating that there is no significant effect or relationship in the data.
• Alternative Hypothesis (Ha): Contradicts the null hypothesis and proposes a specific effect or relationship that researchers want to investigate.
• Nondirectional Hypothesis: An alternative hypothesis that doesn't specify the direction of the effect, leaving it open for both positive and negative possibilities.

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About the author.

Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football.

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Hypothesis Testing Calculator

Related: confidence interval calculator, type ii error.

The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.

Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.

To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.

In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.

To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.

When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.

Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.

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Hypothesis Testing

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CO-6: Apply basic concepts of probability, random variation, and commonly used statistical probability distributions.

Learning Objectives

LO 6.26: Outline the logic and process of hypothesis testing.

LO 6.27: Explain what the p-value is and how it is used to draw conclusions.

Video: Hypothesis Testing (8:43)

Introduction

We are in the middle of the part of the course that has to do with inference for one variable.

So far, we talked about point estimation and learned how interval estimation enhances it by quantifying the magnitude of the estimation error (with a certain level of confidence) in the form of the margin of error. The result is the confidence interval — an interval that, with a certain confidence, we believe captures the unknown parameter.

We are now moving to the other kind of inference, hypothesis testing . We say that hypothesis testing is “the other kind” because, unlike the inferential methods we presented so far, where the goal was estimating the unknown parameter, the idea, logic and goal of hypothesis testing are quite different.

In the first two parts of this section we will discuss the idea behind hypothesis testing, explain how it works, and introduce new terminology that emerges in this form of inference. The final two parts will be more specific and will discuss hypothesis testing for the population proportion ( p ) and the population mean ( μ, mu).

If this is your first statistics course, you will need to spend considerable time on this topic as there are many new ideas. Many students find this process and its logic difficult to understand in the beginning.

In this section, we will use the hypothesis test for a population proportion to motivate our understanding of the process. We will conduct these tests manually. For all future hypothesis test procedures, including problems involving means, we will use software to obtain the results and focus on interpreting them in the context of our scenario.

General Idea and Logic of Hypothesis Testing

The purpose of this section is to gradually build your understanding about how statistical hypothesis testing works. We start by explaining the general logic behind the process of hypothesis testing. Once we are confident that you understand this logic, we will add some more details and terminology.

To start our discussion about the idea behind statistical hypothesis testing, consider the following example:

A case of suspected cheating on an exam is brought in front of the disciplinary committee at a certain university.

There are two opposing claims in this case:

• The student’s claim: I did not cheat on the exam.
• The instructor’s claim: The student did cheat on the exam.

Adhering to the principle “innocent until proven guilty,” the committee asks the instructor for evidence to support his claim. The instructor explains that the exam had two versions, and shows the committee members that on three separate exam questions, the student used in his solution numbers that were given in the other version of the exam.

The committee members all agree that it would be extremely unlikely to get evidence like that if the student’s claim of not cheating had been true. In other words, the committee members all agree that the instructor brought forward strong enough evidence to reject the student’s claim, and conclude that the student did cheat on the exam.

What does this example have to do with statistics?

While it is true that this story seems unrelated to statistics, it captures all the elements of hypothesis testing and the logic behind it. Before you read on to understand why, it would be useful to read the example again. Please do so now.

Statistical hypothesis testing is defined as:

• Assessing evidence provided by the data against the null claim (the claim which is to be assumed true unless enough evidence exists to reject it).

Here is how the process of statistical hypothesis testing works:

• We have two claims about what is going on in the population. Let’s call them claim 1 (this will be the null claim or hypothesis) and claim 2 (this will be the alternative) . Much like the story above, where the student’s claim is challenged by the instructor’s claim, the null claim 1 is challenged by the alternative claim 2. (For us, these claims are usually about the value of population parameter(s) or about the existence or nonexistence of a relationship between two variables in the population).
• We choose a sample, collect relevant data and summarize them (this is similar to the instructor collecting evidence from the student’s exam). For statistical tests, this step will also involve checking any conditions or assumptions.
• We figure out how likely it is to observe data like the data we obtained, if claim 1 is true. (Note that the wording “how likely …” implies that this step requires some kind of probability calculation). In the story, the committee members assessed how likely it is to observe evidence such as the instructor provided, had the student’s claim of not cheating been true.
• If, after assuming claim 1 is true, we find that it would be extremely unlikely to observe data as strong as ours or stronger in favor of claim 2, then we have strong evidence against claim 1, and we reject it in favor of claim 2. Later we will see this corresponds to a small p-value.
• If, after assuming claim 1 is true, we find that observing data as strong as ours or stronger in favor of claim 2 is NOT VERY UNLIKELY , then we do not have enough evidence against claim 1, and therefore we cannot reject it in favor of claim 2. Later we will see this corresponds to a p-value which is not small.

In our story, the committee decided that it would be extremely unlikely to find the evidence that the instructor provided had the student’s claim of not cheating been true. In other words, the members felt that it is extremely unlikely that it is just a coincidence (random chance) that the student used the numbers from the other version of the exam on three separate problems. The committee members therefore decided to reject the student’s claim and concluded that the student had, indeed, cheated on the exam. (Wouldn’t you conclude the same?)

Hopefully this example helped you understand the logic behind hypothesis testing.

Interactive Applet: Reasoning of a Statistical Test

To strengthen your understanding of the process of hypothesis testing and the logic behind it, let’s look at three statistical examples.

A recent study estimated that 20% of all college students in the United States smoke. The head of Health Services at Goodheart University (GU) suspects that the proportion of smokers may be lower at GU. In hopes of confirming her claim, the head of Health Services chooses a random sample of 400 Goodheart students, and finds that 70 of them are smokers.

Let’s analyze this example using the 4 steps outlined above:

• claim 1: The proportion of smokers at Goodheart is 0.20.
• claim 2: The proportion of smokers at Goodheart is less than 0.20.

Claim 1 basically says “nothing special goes on at Goodheart University; the proportion of smokers there is no different from the proportion in the entire country.” This claim is challenged by the head of Health Services, who suspects that the proportion of smokers at Goodheart is lower.

• Choosing a sample and collecting data: A sample of n = 400 was chosen, and summarizing the data revealed that the sample proportion of smokers is p -hat = 70/400 = 0.175.While it is true that 0.175 is less than 0.20, it is not clear whether this is strong enough evidence against claim 1. We must account for sampling variation.
• Assessment of evidence: In order to assess whether the data provide strong enough evidence against claim 1, we need to ask ourselves: How surprising is it to get a sample proportion as low as p -hat = 0.175 (or lower), assuming claim 1 is true? In other words, we need to find how likely it is that in a random sample of size n = 400 taken from a population where the proportion of smokers is p = 0.20 we’ll get a sample proportion as low as p -hat = 0.175 (or lower).It turns out that the probability that we’ll get a sample proportion as low as p -hat = 0.175 (or lower) in such a sample is roughly 0.106 (do not worry about how this was calculated at this point – however, if you think about it hopefully you can see that the key is the sampling distribution of p -hat).
• Conclusion: Well, we found that if claim 1 were true there is a probability of 0.106 of observing data like that observed or more extreme. Now you have to decide …Do you think that a probability of 0.106 makes our data rare enough (surprising enough) under claim 1 so that the fact that we did observe it is enough evidence to reject claim 1? Or do you feel that a probability of 0.106 means that data like we observed are not very likely when claim 1 is true, but they are not unlikely enough to conclude that getting such data is sufficient evidence to reject claim 1. Basically, this is your decision. However, it would be nice to have some kind of guideline about what is generally considered surprising enough.

A certain prescription allergy medicine is supposed to contain an average of 245 parts per million (ppm) of a certain chemical. If the concentration is higher than 245 ppm, the drug will likely cause unpleasant side effects, and if the concentration is below 245 ppm, the drug may be ineffective. The manufacturer wants to check whether the mean concentration in a large shipment is the required 245 ppm or not. To this end, a random sample of 64 portions from the large shipment is tested, and it is found that the sample mean concentration is 250 ppm with a sample standard deviation of 12 ppm.

• Claim 1: The mean concentration in the shipment is the required 245 ppm.
• Claim 2: The mean concentration in the shipment is not the required 245 ppm.

Note that again, claim 1 basically says: “There is nothing unusual about this shipment, the mean concentration is the required 245 ppm.” This claim is challenged by the manufacturer, who wants to check whether that is, indeed, the case or not.

• Choosing a sample and collecting data: A sample of n = 64 portions is chosen and after summarizing the data it is found that the sample mean concentration is x-bar = 250 and the sample standard deviation is s = 12.Is the fact that x-bar = 250 is different from 245 strong enough evidence to reject claim 1 and conclude that the mean concentration in the whole shipment is not the required 245? In other words, do the data provide strong enough evidence to reject claim 1?
• Assessing the evidence: In order to assess whether the data provide strong enough evidence against claim 1, we need to ask ourselves the following question: If the mean concentration in the whole shipment were really the required 245 ppm (i.e., if claim 1 were true), how surprising would it be to observe a sample of 64 portions where the sample mean concentration is off by 5 ppm or more (as we did)? It turns out that it would be extremely unlikely to get such a result if the mean concentration were really the required 245. There is only a probability of 0.0007 (i.e., 7 in 10,000) of that happening. (Do not worry about how this was calculated at this point, but again, the key will be the sampling distribution.)
• Making conclusions: Here, it is pretty clear that a sample like the one we observed or more extreme is VERY rare (or extremely unlikely) if the mean concentration in the shipment were really the required 245 ppm. The fact that we did observe such a sample therefore provides strong evidence against claim 1, so we reject it and conclude with very little doubt that the mean concentration in the shipment is not the required 245 ppm.

Do you think that you’re getting it? Let’s make sure, and look at another example.

Is there a relationship between gender and combined scores (Math + Verbal) on the SAT exam?

Following a report on the College Board website, which showed that in 2003, males scored generally higher than females on the SAT exam, an educational researcher wanted to check whether this was also the case in her school district. The researcher chose random samples of 150 males and 150 females from her school district, collected data on their SAT performance and found the following:

Again, let’s see how the process of hypothesis testing works for this example:

• Claim 1: Performance on the SAT is not related to gender (males and females score the same).
• Claim 2: Performance on the SAT is related to gender – males score higher.

Note that again, claim 1 basically says: “There is nothing going on between the variables SAT and gender.” Claim 2 represents what the researcher wants to check, or suspects might actually be the case.

• Choosing a sample and collecting data: Data were collected and summarized as given above. Is the fact that the sample mean score of males (1,025) is higher than the sample mean score of females (1,010) by 15 points strong enough information to reject claim 1 and conclude that in this researcher’s school district, males score higher on the SAT than females?
• Assessment of evidence: In order to assess whether the data provide strong enough evidence against claim 1, we need to ask ourselves: If SAT scores are in fact not related to gender (claim 1 is true), how likely is it to get data like the data we observed, in which the difference between the males’ average and females’ average score is as high as 15 points or higher? It turns out that the probability of observing such a sample result if SAT score is not related to gender is approximately 0.29 (Again, do not worry about how this was calculated at this point).
• Conclusion: Here, we have an example where observing a sample like the one we observed or more extreme is definitely not surprising (roughly 30% chance) if claim 1 were true (i.e., if indeed there is no difference in SAT scores between males and females). We therefore conclude that our data does not provide enough evidence for rejecting claim 1.
• “The data provide enough evidence to reject claim 1 and accept claim 2”; or
• “The data do not provide enough evidence to reject claim 1.”

In particular, note that in the second type of conclusion we did not say: “ I accept claim 1 ,” but only “ I don’t have enough evidence to reject claim 1 .” We will come back to this issue later, but this is a good place to make you aware of this subtle difference.

Hopefully by now, you understand the logic behind the statistical hypothesis testing process. Here is a summary:

Learn by Doing: Logic of Hypothesis Testing

Did I Get This?: Logic of Hypothesis Testing

Steps in Hypothesis Testing

Video: Steps in Hypothesis Testing (16:02)

Now that we understand the general idea of how statistical hypothesis testing works, let’s go back to each of the steps and delve slightly deeper, getting more details and learning some terminology.

Hypothesis Testing Step 1: State the Hypotheses

In all three examples, our aim is to decide between two opposing points of view, Claim 1 and Claim 2. In hypothesis testing, Claim 1 is called the null hypothesis (denoted “ Ho “), and Claim 2 plays the role of the alternative hypothesis (denoted “ Ha “). As we saw in the three examples, the null hypothesis suggests nothing special is going on; in other words, there is no change from the status quo, no difference from the traditional state of affairs, no relationship. In contrast, the alternative hypothesis disagrees with this, stating that something is going on, or there is a change from the status quo, or there is a difference from the traditional state of affairs. The alternative hypothesis, Ha, usually represents what we want to check or what we suspect is really going on.

Let’s go back to our three examples and apply the new notation:

In example 1:

• Ho: The proportion of smokers at GU is 0.20.
• Ha: The proportion of smokers at GU is less than 0.20.

In example 2:

• Ho: The mean concentration in the shipment is the required 245 ppm.
• Ha: The mean concentration in the shipment is not the required 245 ppm.

In example 3:

• Ho: Performance on the SAT is not related to gender (males and females score the same).
• Ha: Performance on the SAT is related to gender – males score higher.

Learn by Doing: State the Hypotheses

Did I Get This?: State the Hypotheses

Hypothesis Testing Step 2: Collect Data, Check Conditions and Summarize Data

This step is pretty obvious. This is what inference is all about. You look at sampled data in order to draw conclusions about the entire population. In the case of hypothesis testing, based on the data, you draw conclusions about whether or not there is enough evidence to reject Ho.

There is, however, one detail that we would like to add here. In this step we collect data and summarize it. Go back and look at the second step in our three examples. Note that in order to summarize the data we used simple sample statistics such as the sample proportion ( p -hat), sample mean (x-bar) and the sample standard deviation (s).

In practice, you go a step further and use these sample statistics to summarize the data with what’s called a test statistic . We are not going to go into any details right now, but we will discuss test statistics when we go through the specific tests.

This step will also involve checking any conditions or assumptions required to use the test.

Hypothesis Testing Step 3: Assess the Evidence

As we saw, this is the step where we calculate how likely is it to get data like that observed (or more extreme) when Ho is true. In a sense, this is the heart of the process, since we draw our conclusions based on this probability.

• If this probability is very small (see example 2), then that means that it would be very surprising to get data like that observed (or more extreme) if Ho were true. The fact that we did observe such data is therefore evidence against Ho, and we should reject it.
• On the other hand, if this probability is not very small (see example 3) this means that observing data like that observed (or more extreme) is not very surprising if Ho were true. The fact that we observed such data does not provide evidence against Ho. This crucial probability, therefore, has a special name. It is called the p-value of the test.

In our three examples, the p-values were given to you (and you were reassured that you didn’t need to worry about how these were derived yet):

• Example 1: p-value = 0.106
• Example 2: p-value = 0.0007
• Example 3: p-value = 0.29

Obviously, the smaller the p-value, the more surprising it is to get data like ours (or more extreme) when Ho is true, and therefore, the stronger the evidence the data provide against Ho.

Looking at the three p-values of our three examples, we see that the data that we observed in example 2 provide the strongest evidence against the null hypothesis, followed by example 1, while the data in example 3 provides the least evidence against Ho.

• Right now we will not go into specific details about p-value calculations, but just mention that since the p-value is the probability of getting data like those observed (or more extreme) when Ho is true, it would make sense that the calculation of the p-value will be based on the data summary, which, as we mentioned, is the test statistic. Indeed, this is the case. In practice, we will mostly use software to provide the p-value for us.

Hypothesis Testing Step 4: Making Conclusions

Since our statistical conclusion is based on how small the p-value is, or in other words, how surprising our data are when Ho is true, it would be nice to have some kind of guideline or cutoff that will help determine how small the p-value must be, or how “rare” (unlikely) our data must be when Ho is true, for us to conclude that we have enough evidence to reject Ho.

This cutoff exists, and because it is so important, it has a special name. It is called the significance level of the test and is usually denoted by the Greek letter α (alpha). The most commonly used significance level is α (alpha) = 0.05 (or 5%). This means that:

• if the p-value < α (alpha) (usually 0.05), then the data we obtained is considered to be “rare (or surprising) enough” under the assumption that Ho is true, and we say that the data provide statistically significant evidence against Ho, so we reject Ho and thus accept Ha.
• if the p-value > α (alpha)(usually 0.05), then our data are not considered to be “surprising enough” under the assumption that Ho is true, and we say that our data do not provide enough evidence to reject Ho (or, equivalently, that the data do not provide enough evidence to accept Ha).

Now that we have a cutoff to use, here are the appropriate conclusions for each of our examples based upon the p-values we were given.

In Example 1:

• Using our cutoff of 0.05, we fail to reject Ho.
• Conclusion : There IS NOT enough evidence that the proportion of smokers at GU is less than 0.20
• Still we should consider: Does the evidence seen in the data provide any practical evidence towards our alternative hypothesis?

In Example 2:

• Using our cutoff of 0.05, we reject Ho.
• Conclusion : There IS enough evidence that the mean concentration in the shipment is not the required 245 ppm.

In Example 3:

• Conclusion : There IS NOT enough evidence that males score higher on average than females on the SAT.

Notice that all of the above conclusions are written in terms of the alternative hypothesis and are given in the context of the situation. In no situation have we claimed the null hypothesis is true. Be very careful of this and other issues discussed in the following comments.

• Although the significance level provides a good guideline for drawing our conclusions, it should not be treated as an incontrovertible truth. There is a lot of room for personal interpretation. What if your p-value is 0.052? You might want to stick to the rules and say “0.052 > 0.05 and therefore I don’t have enough evidence to reject Ho”, but you might decide that 0.052 is small enough for you to believe that Ho should be rejected. It should be noted that scientific journals do consider 0.05 to be the cutoff point for which any p-value below the cutoff indicates enough evidence against Ho, and any p-value above it, or even equal to it , indicates there is not enough evidence against Ho. Although a p-value between 0.05 and 0.10 is often reported as marginally statistically significant.
• It is important to draw your conclusions in context . It is never enough to say: “p-value = …, and therefore I have enough evidence to reject Ho at the 0.05 significance level.” You should always word your conclusion in terms of the data. Although we will use the terminology of “rejecting Ho” or “failing to reject Ho” – this is mostly due to the fact that we are instructing you in these concepts. In practice, this language is rarely used. We also suggest writing your conclusion in terms of the alternative hypothesis.Is there or is there not enough evidence that the alternative hypothesis is true?
• Let’s go back to the issue of the nature of the two types of conclusions that I can make.
• Either I reject Ho (when the p-value is smaller than the significance level)
• or I cannot reject Ho (when the p-value is larger than the significance level).

As we mentioned earlier, note that the second conclusion does not imply that I accept Ho, but just that I don’t have enough evidence to reject it. Saying (by mistake) “I don’t have enough evidence to reject Ho so I accept it” indicates that the data provide evidence that Ho is true, which is not necessarily the case . Consider the following slightly artificial yet effective example:

An employer claims to subscribe to an “equal opportunity” policy, not hiring men any more often than women for managerial positions. Is this credible? You’re not sure, so you want to test the following two hypotheses:

• Ho: The proportion of male managers hired is 0.5
• Ha: The proportion of male managers hired is more than 0.5

Data: You choose at random three of the new managers who were hired in the last 5 years and find that all 3 are men.

Assessing Evidence: If the proportion of male managers hired is really 0.5 (Ho is true), then the probability that the random selection of three managers will yield three males is therefore 0.5 * 0.5 * 0.5 = 0.125. This is the p-value (using the multiplication rule for independent events).

Conclusion: Using 0.05 as the significance level, you conclude that since the p-value = 0.125 > 0.05, the fact that the three randomly selected managers were all males is not enough evidence to reject the employer’s claim of subscribing to an equal opportunity policy (Ho).

However, the data (all three selected are males) definitely does NOT provide evidence to accept the employer’s claim (Ho).

Learn By Doing: Using p-values

Did I Get This?: Using p-values

Comment about wording: Another common wording in scientific journals is:

• “The results are statistically significant” – when the p-value < α (alpha).
• “The results are not statistically significant” – when the p-value > α (alpha).

Often you will see significance levels reported with additional description to indicate the degree of statistical significance. A general guideline (although not required in our course) is:

• If 0.01 ≤ p-value < 0.05, then the results are (statistically) significant .
• If 0.001 ≤ p-value < 0.01, then the results are highly statistically significant .
• If p-value < 0.001, then the results are very highly statistically significant .
• If p-value > 0.05, then the results are not statistically significant (NS).
• If 0.05 ≤ p-value < 0.10, then the results are marginally statistically significant .

Let’s summarize

We learned quite a lot about hypothesis testing. We learned the logic behind it, what the key elements are, and what types of conclusions we can and cannot draw in hypothesis testing. Here is a quick recap:

Video: Hypothesis Testing Overview (2:20)

Here are a few more activities if you need some additional practice.

Did I Get This?: Hypothesis Testing Overview

• Notice that the p-value is an example of a conditional probability . We calculate the probability of obtaining results like those of our data (or more extreme) GIVEN the null hypothesis is true. We could write P(Obtaining results like ours or more extreme | Ho is True).
• We could write P(Obtaining a test statistic as or more extreme than ours | Ho is True).
• In this case we are asking “Assuming the null hypothesis is true, how rare is it to observe something as or more extreme than what I have found in my data?”
• If after assuming the null hypothesis is true, what we have found in our data is extremely rare (small p-value), this provides evidence to reject our assumption that Ho is true in favor of Ha.
• The p-value can also be thought of as the probability, assuming the null hypothesis is true, that the result we have seen is solely due to random error (or random chance). We have already seen that statistics from samples collected from a population vary. There is random error or random chance involved when we sample from populations.

In this setting, if the p-value is very small, this implies, assuming the null hypothesis is true, that it is extremely unlikely that the results we have obtained would have happened due to random error alone, and thus our assumption (Ho) is rejected in favor of the alternative hypothesis (Ha).

• It is EXTREMELY important that you find a definition of the p-value which makes sense to you. New students often need to contemplate this idea repeatedly through a variety of examples and explanations before becoming comfortable with this idea. It is one of the two most important concepts in statistics (the other being confidence intervals).
• We infer that the alternative hypothesis is true ONLY by rejecting the null hypothesis.
• A statistically significant result is one that has a very low probability of occurring if the null hypothesis is true.
• Results which are statistically significant may or may not have practical significance and vice versa.

Error and Power

LO 6.28: Define a Type I and Type II error in general and in the context of specific scenarios.

LO 6.29: Explain the concept of the power of a statistical test including the relationship between power, sample size, and effect size.

Video: Errors and Power (12:03)

Type I and Type II Errors in Hypothesis Tests

We have not yet discussed the fact that we are not guaranteed to make the correct decision by this process of hypothesis testing. Maybe you are beginning to see that there is always some level of uncertainty in statistics.

Let’s think about what we know already and define the possible errors we can make in hypothesis testing. When we conduct a hypothesis test, we choose one of two possible conclusions based upon our data.

If the p-value is smaller than your pre-specified significance level (α, alpha), you reject the null hypothesis and either

• You have made the correct decision since the null hypothesis is false
• You have made an error ( Type I ) and rejected Ho when in fact Ho is true (your data happened to be a RARE EVENT under Ho)

If the p-value is greater than (or equal to) your chosen significance level (α, alpha), you fail to reject the null hypothesis and either

• You have made the correct decision since the null hypothesis is true
• You have made an error ( Type II ) and failed to reject Ho when in fact Ho is false (the alternative hypothesis, Ha, is true)

The following summarizes the four possible results which can be obtained from a hypothesis test. Notice the rows represent the decision made in the hypothesis test and the columns represent the (usually unknown) truth in reality.

Although the truth is unknown in practice – or we would not be conducting the test – we know it must be the case that either the null hypothesis is true or the null hypothesis is false. It is also the case that either decision we make in a hypothesis test can result in an incorrect conclusion!

A TYPE I Error occurs when we Reject Ho when, in fact, Ho is True. In this case, we mistakenly reject a true null hypothesis.

• P(TYPE I Error) = P(Reject Ho | Ho is True) = α = alpha = Significance Level

A TYPE II Error occurs when we fail to Reject Ho when, in fact, Ho is False. In this case we fail to reject a false null hypothesis.

P(TYPE II Error) = P(Fail to Reject Ho | Ho is False) = β = beta

When our significance level is 5%, we are saying that we will allow ourselves to make a Type I error less than 5% of the time. In the long run, if we repeat the process, 5% of the time we will find a p-value < 0.05 when in fact the null hypothesis was true.

In this case, our data represent a rare occurrence which is unlikely to happen but is still possible. For example, suppose we toss a coin 10 times and obtain 10 heads, this is unlikely for a fair coin but not impossible. We might conclude the coin is unfair when in fact we simply saw a very rare event for this fair coin.

Our testing procedure CONTROLS for the Type I error when we set a pre-determined value for the significance level.

Notice that these probabilities are conditional probabilities. This is one more reason why conditional probability is an important concept in statistics.

Unfortunately, calculating the probability of a Type II error requires us to know the truth about the population. In practice we can only calculate this probability using a series of “what if” calculations which depend upon the type of problem.

Comment: As you initially read through the examples below, focus on the broad concepts instead of the small details. It is not important to understand how to calculate these values yourself at this point.

• Try to understand the pictures we present. Which pictures represent an assumed null hypothesis and which represent an alternative?
• It may be useful to come back to this page (and the activities here) after you have reviewed the rest of the section on hypothesis testing and have worked a few problems yourself.

Interactive Applet: Statistical Significance

Here are two examples of using an older version of this applet. It looks slightly different but the same settings and options are available in the version above.

In both cases we will consider IQ scores.

Our null hypothesis is that the true mean is 100. Assume the standard deviation is 16 and we will specify a significance level of 5%.

In this example we will specify that the true mean is indeed 100 so that the null hypothesis is true. Most of the time (95%), when we generate a sample, we should fail to reject the null hypothesis since the null hypothesis is indeed true.

Here is one sample that results in a correct decision:

In the sample above, we obtain an x-bar of 105, which is drawn on the distribution which assumes μ (mu) = 100 (the null hypothesis is true). Notice the sample is shown as blue dots along the x-axis and the shaded region shows for which values of x-bar we would reject the null hypothesis. In other words, we would reject Ho whenever the x-bar falls in the shaded region.

Enter the same values and generate samples until you obtain a Type I error (you falsely reject the null hypothesis). You should see something like this:

If you were to generate 100 samples, you should have around 5% where you rejected Ho. These would be samples which would result in a Type I error.

The previous example illustrates a correct decision and a Type I error when the null hypothesis is true. The next example illustrates a correct decision and Type II error when the null hypothesis is false. In this case, we must specify the true population mean.

Let’s suppose we are sampling from an honors program and that the true mean IQ for this population is 110. We do not know the probability of a Type II error without more detailed calculations.

Let’s start with a sample which results in a correct decision.

In the sample above, we obtain an x-bar of 111, which is drawn on the distribution which assumes μ (mu) = 100 (the null hypothesis is true).

Enter the same values and generate samples until you obtain a Type II error (you fail to reject the null hypothesis). You should see something like this:

You should notice that in this case (when Ho is false), it is easier to obtain an incorrect decision (a Type II error) than it was in the case where Ho is true. If you generate 100 samples, you can approximate the probability of a Type II error.

We can find the probability of a Type II error by visualizing both the assumed distribution and the true distribution together. The image below is adapted from an applet we will use when we discuss the power of a statistical test.

There is a 37.4% chance that, in the long run, we will make a Type II error and fail to reject the null hypothesis when in fact the true mean IQ is 110 in the population from which we sample our 10 individuals.

Can you visualize what will happen if the true population mean is really 115 or 108? When will the Type II error increase? When will it decrease? We will look at this idea again when we discuss the concept of power in hypothesis tests.

• It is important to note that there is a trade-off between the probability of a Type I and a Type II error. If we decrease the probability of one of these errors, the probability of the other will increase! The practical result of this is that if we require stronger evidence to reject the null hypothesis (smaller significance level = probability of a Type I error), we will increase the chance that we will be unable to reject the null hypothesis when in fact Ho is false (increases the probability of a Type II error).
• When α (alpha) = 0.05 we obtained a Type II error probability of 0.374 = β = beta

• When α (alpha) = 0.01 (smaller than before) we obtain a Type II error probability of 0.644 = β = beta (larger than before)

• As the blue line in the picture moves farther right, the significance level (α, alpha) is decreasing and the Type II error probability is increasing.
• As the blue line in the picture moves farther left, the significance level (α, alpha) is increasing and the Type II error probability is decreasing

Let’s return to our very first example and define these two errors in context.

• Ho = The student’s claim: I did not cheat on the exam.
• Ha = The instructor’s claim: The student did cheat on the exam.

Adhering to the principle “innocent until proven guilty,” the committee asks the instructor for evidence to support his claim.

There are four possible outcomes of this process. There are two possible correct decisions:

• The student did cheat on the exam and the instructor brings enough evidence to reject Ho and conclude the student did cheat on the exam. This is a CORRECT decision!
• The student did not cheat on the exam and the instructor fails to provide enough evidence that the student did cheat on the exam. This is a CORRECT decision!

Both the correct decisions and the possible errors are fairly easy to understand but with the errors, you must be careful to identify and define the two types correctly.

TYPE I Error: Reject Ho when Ho is True

• The student did not cheat on the exam but the instructor brings enough evidence to reject Ho and conclude the student cheated on the exam. This is a Type I Error.

TYPE II Error: Fail to Reject Ho when Ho is False

• The student did cheat on the exam but the instructor fails to provide enough evidence that the student cheated on the exam. This is a Type II Error.

In most situations, including this one, it is more “acceptable” to have a Type II error than a Type I error. Although allowing a student who cheats to go unpunished might be considered a very bad problem, punishing a student for something he or she did not do is usually considered to be a more severe error. This is one reason we control for our Type I error in the process of hypothesis testing.

Did I Get This?: Type I and Type II Errors (in context)

• The probabilities of Type I and Type II errors are closely related to the concepts of sensitivity and specificity that we discussed previously. Consider the following hypotheses:

Ho: The individual does not have diabetes (status quo, nothing special happening)

Ha: The individual does have diabetes (something is going on here)

In this setting:

When someone tests positive for diabetes we would reject the null hypothesis and conclude the person has diabetes (we may or may not be correct!).

When someone tests negative for diabetes we would fail to reject the null hypothesis so that we fail to conclude the person has diabetes (we may or may not be correct!)

Let’s take it one step further:

Sensitivity = P(Test + | Have Disease) which in this setting equals P(Reject Ho | Ho is False) = 1 – P(Fail to Reject Ho | Ho is False) = 1 – β = 1 – beta

Specificity = P(Test – | No Disease) which in this setting equals P(Fail to Reject Ho | Ho is True) = 1 – P(Reject Ho | Ho is True) = 1 – α = 1 – alpha

Notice that sensitivity and specificity relate to the probability of making a correct decision whereas α (alpha) and β (beta) relate to the probability of making an incorrect decision.

Usually α (alpha) = 0.05 so that the specificity listed above is 0.95 or 95%.

Next, we will see that the sensitivity listed above is the power of the hypothesis test!

Reasons for a Type I Error in Practice

Assuming that you have obtained a quality sample:

• The reason for a Type I error is random chance.
• When a Type I error occurs, our observed data represented a rare event which indicated evidence in favor of the alternative hypothesis even though the null hypothesis was actually true.

Reasons for a Type II Error in Practice

Again, assuming that you have obtained a quality sample, now we have a few possibilities depending upon the true difference that exists.

• The sample size is too small to detect an important difference. This is the worst case, you should have obtained a larger sample. In this situation, you may notice that the effect seen in the sample seems PRACTICALLY significant and yet the p-value is not small enough to reject the null hypothesis.
• The sample size is reasonable for the important difference but the true difference (which might be somewhat meaningful or interesting) is smaller than your test was capable of detecting. This is tolerable as you were not interested in being able to detect this difference when you began your study. In this situation, you may notice that the effect seen in the sample seems to have some potential for practical significance.
• The sample size is more than adequate, the difference that was not detected is meaningless in practice. This is not a problem at all and is in effect a “correct decision” since the difference you did not detect would have no practical meaning.
• Note: We will discuss the idea of practical significance later in more detail.

Power of a Hypothesis Test

It is often the case that we truly wish to prove the alternative hypothesis. It is reasonable that we would be interested in the probability of correctly rejecting the null hypothesis. In other words, the probability of rejecting the null hypothesis, when in fact the null hypothesis is false. This can also be thought of as the probability of being able to detect a (pre-specified) difference of interest to the researcher.

Let’s begin with a realistic example of how power can be described in a study.

In a clinical trial to study two medications for weight loss, we have an 80% chance to detect a difference in the weight loss between the two medications of 10 pounds. In other words, the power of the hypothesis test we will conduct is 80%.

In other words, if one medication comes from a population with an average weight loss of 25 pounds and the other comes from a population with an average weight loss of 15 pounds, we will have an 80% chance to detect that difference using the sample we have in our trial.

If we were to repeat this trial many times, 80% of the time we will be able to reject the null hypothesis (that there is no difference between the medications) and 20% of the time we will fail to reject the null hypothesis (and make a Type II error!).

The difference of 10 pounds in the previous example, is often called the effect size . The measure of the effect differs depending on the particular test you are conducting but is always some measure related to the true effect in the population. In this example, it is the difference between two population means.

Recall the definition of a Type II error:

Notice that P(Reject Ho | Ho is False) = 1 – P(Fail to Reject Ho | Ho is False) = 1 – β = 1- beta.

The POWER of a hypothesis test is the probability of rejecting the null hypothesis when the null hypothesis is false . This can also be stated as the probability of correctly rejecting the null hypothesis .

POWER = P(Reject Ho | Ho is False) = 1 – β = 1 – beta

Power is the test’s ability to correctly reject the null hypothesis. A test with high power has a good chance of being able to detect the difference of interest to us, if it exists .

As we mentioned on the bottom of the previous page, this can be thought of as the sensitivity of the hypothesis test if you imagine Ho = No disease and Ha = Disease.

Factors Affecting the Power of a Hypothesis Test

The power of a hypothesis test is affected by numerous quantities (similar to the margin of error in a confidence interval).

Assume that the null hypothesis is false for a given hypothesis test. All else being equal, we have the following:

• Larger samples result in a greater chance to reject the null hypothesis which means an increase in the power of the hypothesis test.
• If the effect size is larger, it will become easier for us to detect. This results in a greater chance to reject the null hypothesis which means an increase in the power of the hypothesis test. The effect size varies for each test and is usually closely related to the difference between the hypothesized value and the true value of the parameter under study.
• From the relationship between the probability of a Type I and a Type II error (as α (alpha) decreases, β (beta) increases), we can see that as α (alpha) decreases, Power = 1 – β = 1 – beta also decreases.
• There are other mathematical ways to change the power of a hypothesis test, such as changing the population standard deviation; however, these are not quantities that we can usually control so we will not discuss them here.

In practice, we specify a significance level and a desired power to detect a difference which will have practical meaning to us and this determines the sample size required for the experiment or study.

For most grants involving statistical analysis, power calculations must be completed to illustrate that the study will have a reasonable chance to detect an important effect. Otherwise, the money spent on the study could be wasted. The goal is usually to have a power close to 80%.

For example, if there is only a 5% chance to detect an important difference between two treatments in a clinical trial, this would result in a waste of time, effort, and money on the study since, when the alternative hypothesis is true, the chance a treatment effect can be found is very small.

• In order to calculate the power of a hypothesis test, we must specify the “truth.” As we mentioned previously when discussing Type II errors, in practice we can only calculate this probability using a series of “what if” calculations which depend upon the type of problem.

The following activity involves working with an interactive applet to study power more carefully.

Learn by Doing: Power of Hypothesis Tests

The following reading is an excellent discussion about Type I and Type II errors.

(Optional) Outside Reading: A Good Discussion of Power (≈ 2500 words)

We will not be asking you to perform power calculations manually. You may be asked to use online calculators and applets. Most statistical software packages offer some ability to complete power calculations. There are also many online calculators for power and sample size on the internet, for example, Russ Lenth’s power and sample-size page .

Proportions (Introduction & Step 1)

CO-4: Distinguish among different measurement scales, choose the appropriate descriptive and inferential statistical methods based on these distinctions, and interpret the results.

LO 4.33: In a given context, distinguish between situations involving a population proportion and a population mean and specify the correct null and alternative hypothesis for the scenario.

LO 4.34: Carry out a complete hypothesis test for a population proportion by hand.

Video: Proportions (Introduction & Step 1) (7:18)

Now that we understand the process of hypothesis testing and the logic behind it, we are ready to start learning about specific statistical tests (also known as significance tests).

The first test we are going to learn is the test about the population proportion (p).

This test is widely known as the “z-test for the population proportion (p).”

We will understand later where the “z-test” part is coming from.

This will be the only type of problem you will complete entirely “by-hand” in this course. Our goal is to use this example to give you the tools you need to understand how this process works. After working a few problems, you should review the earlier material again. You will likely need to review the terminology and concepts a few times before you fully understand the process.

In reality, you will often be conducting more complex statistical tests and allowing software to provide the p-value. In these settings it will be important to know what test to apply for a given situation and to be able to explain the results in context.

Review: Types of Variables

When we conduct a test about a population proportion, we are working with a categorical variable. Later in the course, after we have learned a variety of hypothesis tests, we will need to be able to identify which test is appropriate for which situation. Identifying the variable as categorical or quantitative is an important component of choosing an appropriate hypothesis test.

Learn by Doing: Review Types of Variables

One Sample Z-Test for a Population Proportion

In this part of our discussion on hypothesis testing, we will go into details that we did not go into before. More specifically, we will use this test to introduce the idea of a test statistic , and details about how p-values are calculated .

Let’s start by introducing the three examples, which will be the leading examples in our discussion. Each example is followed by a figure illustrating the information provided, as well as the question of interest.

A machine is known to produce 20% defective products, and is therefore sent for repair. After the machine is repaired, 400 products produced by the machine are chosen at random and 64 of them are found to be defective. Do the data provide enough evidence that the proportion of defective products produced by the machine (p) has been reduced as a result of the repair?

The following figure displays the information, as well as the question of interest:

The question of interest helps us formulate the null and alternative hypotheses in terms of p, the proportion of defective products produced by the machine following the repair:

• Ho: p = 0.20 (No change; the repair did not help).
• Ha: p < 0.20 (The repair was effective at reducing the proportion of defective parts).

There are rumors that students at a certain liberal arts college are more inclined to use drugs than U.S. college students in general. Suppose that in a simple random sample of 100 students from the college, 19 admitted to marijuana use. Do the data provide enough evidence to conclude that the proportion of marijuana users among the students in the college (p) is higher than the national proportion, which is 0.157? (This number is reported by the Harvard School of Public Health.)

Again, the following figure displays the information as well as the question of interest:

As before, we can formulate the null and alternative hypotheses in terms of p, the proportion of students in the college who use marijuana:

• Ho: p = 0.157 (same as among all college students in the country).
• Ha: p > 0.157 (higher than the national figure).

Polls on certain topics are conducted routinely in order to monitor changes in the public’s opinions over time. One such topic is the death penalty. In 2003 a poll estimated that 64% of U.S. adults support the death penalty for a person convicted of murder. In a more recent poll, 675 out of 1,000 U.S. adults chosen at random were in favor of the death penalty for convicted murderers. Do the results of this poll provide evidence that the proportion of U.S. adults who support the death penalty for convicted murderers (p) changed between 2003 and the later poll?

Here is a figure that displays the information, as well as the question of interest:

Again, we can formulate the null and alternative hypotheses in term of p, the proportion of U.S. adults who support the death penalty for convicted murderers.

• Ho: p = 0.64 (No change from 2003).
• Ha: p ≠ 0.64 (Some change since 2003).

Learn by Doing: Proportions (Overview)

Did I Get This?: Proportions ( Overview )

Recall that there are basically 4 steps in the process of hypothesis testing:

• STEP 1: State the appropriate null and alternative hypotheses, Ho and Ha.
• STEP 2: Obtain a random sample, collect relevant data, and check whether the data meet the conditions under which the test can be used . If the conditions are met, summarize the data using a test statistic.
• STEP 3: Find the p-value of the test.
• STEP 4: Based on the p-value, decide whether or not the results are statistically significant and draw your conclusions in context.
• Note: In practice, we should always consider the practical significance of the results as well as the statistical significance.

We are now going to go through these steps as they apply to the hypothesis testing for the population proportion p. It should be noted that even though the details will be specific to this particular test, some of the ideas that we will add apply to hypothesis testing in general.

Step 1. Stating the Hypotheses

Here again are the three set of hypotheses that are being tested in each of our three examples:

Has the proportion of defective products been reduced as a result of the repair?

Is the proportion of marijuana users in the college higher than the national figure?

Did the proportion of U.S. adults who support the death penalty change between 2003 and a later poll?

The null hypothesis always takes the form:

• Ho: p = some value

and the alternative hypothesis takes one of the following three forms:

• Ha: p < that value (like in example 1) or
• Ha: p > that value (like in example 2) or
• Ha: p ≠ that value (like in example 3).

Note that it was quite clear from the context which form of the alternative hypothesis would be appropriate. The value that is specified in the null hypothesis is called the null value , and is generally denoted by p 0 . We can say, therefore, that in general the null hypothesis about the population proportion (p) would take the form:

• Ho: p = p 0

We write Ho: p = p 0 to say that we are making the hypothesis that the population proportion has the value of p 0 . In other words, p is the unknown population proportion and p 0 is the number we think p might be for the given situation.

The alternative hypothesis takes one of the following three forms (depending on the context):

Ha: p < p 0 (one-sided)

Ha: p > p 0 (one-sided)

Ha: p ≠ p 0 (two-sided)

The first two possible forms of the alternatives (where the = sign in Ho is challenged by < or >) are called one-sided alternatives , and the third form of alternative (where the = sign in Ho is challenged by ≠) is called a two-sided alternative. To understand the intuition behind these names let’s go back to our examples.

Example 3 (death penalty) is a case where we have a two-sided alternative:

In this case, in order to reject Ho and accept Ha we will need to get a sample proportion of death penalty supporters which is very different from 0.64 in either direction, either much larger or much smaller than 0.64.

In example 2 (marijuana use) we have a one-sided alternative:

Here, in order to reject Ho and accept Ha we will need to get a sample proportion of marijuana users which is much higher than 0.157.

Similarly, in example 1 (defective products), where we are testing:

in order to reject Ho and accept Ha, we will need to get a sample proportion of defective products which is much smaller than 0.20.

Learn by Doing: State Hypotheses (Proportions)

Did I Get This?: State Hypotheses (Proportions)

Proportions (Step 2)

Video: Proportions (Step 2) (12:38)

Step 2. Collect Data, Check Conditions, and Summarize Data

After the hypotheses have been stated, the next step is to obtain a sample (on which the inference will be based), collect relevant data , and summarize them.

It is extremely important that our sample is representative of the population about which we want to draw conclusions. This is ensured when the sample is chosen at random. Beyond the practical issue of ensuring representativeness, choosing a random sample has theoretical importance that we will mention later.

In the case of hypothesis testing for the population proportion (p), we will collect data on the relevant categorical variable from the individuals in the sample and start by calculating the sample proportion p-hat (the natural quantity to calculate when the parameter of interest is p).

Let’s go back to our three examples and add this step to our figures.

As we mentioned earlier without going into details, when we summarize the data in hypothesis testing, we go a step beyond calculating the sample statistic and summarize the data with a test statistic . Every test has a test statistic, which to some degree captures the essence of the test. In fact, the p-value, which so far we have looked upon as “the king” (in the sense that everything is determined by it), is actually determined by (or derived from) the test statistic. We will now introduce the test statistic.

The test statistic is a measure of how far the sample proportion p-hat is from the null value p 0 , the value that the null hypothesis claims is the value of p. In other words, since p-hat is what the data estimates p to be, the test statistic can be viewed as a measure of the “distance” between what the data tells us about p and what the null hypothesis claims p to be.

Let’s use our examples to understand this:

The parameter of interest is p, the proportion of defective products following the repair.

The data estimate p to be p-hat = 0.16

The null hypothesis claims that p = 0.20

The data are therefore 0.04 (or 4 percentage points) below the null hypothesis value.

It is hard to evaluate whether this difference of 4% in defective products is enough evidence to say that the repair was effective at reducing the proportion of defective products, but clearly, the larger the difference, the more evidence it is against the null hypothesis. So if, for example, our sample proportion of defective products had been, say, 0.10 instead of 0.16, then I think you would all agree that cutting the proportion of defective products in half (from 20% to 10%) would be extremely strong evidence that the repair was effective at reducing the proportion of defective products.

The parameter of interest is p, the proportion of students in a college who use marijuana.

The data estimate p to be p-hat = 0.19

The null hypothesis claims that p = 0.157

The data are therefore 0.033 (or 3.3. percentage points) above the null hypothesis value.

The parameter of interest is p, the proportion of U.S. adults who support the death penalty for convicted murderers.

The data estimate p to be p-hat = 0.675

The null hypothesis claims that p = 0.64

There is a difference of 0.035 (or 3.5. percentage points) between the data and the null hypothesis value.

The problem with looking only at the difference between the sample proportion, p-hat, and the null value, p 0 is that we have not taken into account the variability of our estimator p-hat which, as we know from our study of sampling distributions, depends on the sample size.

For this reason, the test statistic cannot simply be the difference between p-hat and p 0 , but must be some form of that formula that accounts for the sample size. In other words, we need to somehow standardize the difference so that comparison between different situations will be possible. We are very close to revealing the test statistic, but before we construct it, let’s be reminded of the following two facts from probability:

Fact 1: When we take a random sample of size n from a population with population proportion p, then

Fact 2: The z-score of any normal value (a value that comes from a normal distribution) is calculated by finding the difference between the value and the mean and then dividing that difference by the standard deviation (of the normal distribution associated with the value). The z-score represents how many standard deviations below or above the mean the value is.

Thus, our test statistic should be a measure of how far the sample proportion p-hat is from the null value p 0 relative to the variation of p-hat (as measured by the standard error of p-hat).

Recall that the standard error is the standard deviation of the sampling distribution for a given statistic. For p-hat, we know the following:

To find the p-value, we will need to determine how surprising our value is assuming the null hypothesis is true. We already have the tools needed for this process from our study of sampling distributions as represented in the table above.

If we assume the null hypothesis is true, we can specify that the center of the distribution of all possible values of p-hat from samples of size 400 would be 0.20 (our null value).

We can calculate the standard error, assuming p = 0.20 as

\(\sqrt{\dfrac{p_{0}\left(1-p_{0}\right)}{n}}=\sqrt{\dfrac{0.2(1-0.2)}{400}}=0.02\)

The following picture represents the sampling distribution of all possible values of p-hat of samples of size 400, assuming the true proportion p is 0.20 and our other requirements for the sampling distribution to be normal are met (we will review these during the next step).

In order to calculate probabilities for the picture above, we would need to find the z-score associated with our result.

This z-score is the test statistic ! In this example, the numerator of our z-score is the difference between p-hat (0.16) and null value (0.20) which we found earlier to be -0.04. The denominator of our z-score is the standard error calculated above (0.02) and thus quickly we find the z-score, our test statistic, to be -2.

The sample proportion based upon this data is 2 standard errors below the null value.

Hopefully you now understand more about the reasons we need probability in statistics!!

Now we will formalize the definition and look at our remaining examples before moving on to the next step, which will be to determine if a normal distribution applies and calculate the p-value.

Test Statistic for Hypothesis Tests for One Proportion is:

\(z=\dfrac{\hat{p}-p_{0}}{\sqrt{\dfrac{p_{0}\left(1-p_{0}\right)}{n}}}\)

It represents the difference between the sample proportion and the null value, measured in standard deviations (standard error of p-hat).

The picture above is a representation of the sampling distribution of p-hat assuming p = p 0 . In other words, this is a model of how p-hat behaves if we are drawing random samples from a population for which Ho is true.

Notice the center of the sampling distribution is at p 0 , which is the hypothesized proportion given in the null hypothesis (Ho: p = p 0 .) We could also mark the axis in standard error units,

\(\sqrt{\dfrac{p_{0}\left(1-p_{0}\right)}{n}}\)

For example, if our null hypothesis claims that the proportion of U.S. adults supporting the death penalty is 0.64, then the sampling distribution is drawn as if the null is true. We draw a normal distribution centered at 0.64 (p 0 ) with a standard error dependent on sample size,

\(\sqrt{\dfrac{0.64(1-0.64)}{n}}\).

Important Comment:

• Note that under the assumption that Ho is true (and if the conditions for the sampling distribution to be normal are satisfied) the test statistic follows a N(0,1) (standard normal) distribution. Another way to say the same thing which is quite common is: “The null distribution of the test statistic is N(0,1).”

By “null distribution,” we mean the distribution under the assumption that Ho is true. As we’ll see and stress again later, the null distribution of the test statistic is what the calculation of the p-value is based on.

Let’s go back to our remaining two examples and find the test statistic in each case:

Since the null hypothesis is Ho: p = 0.157, the standardized (z) score of p-hat = 0.19 is

\(z=\dfrac{0.19-0.157}{\sqrt{\dfrac{0.157(1-0.157)}{100}}} \approx 0.91\)

This is the value of the test statistic for this example.

We interpret this to mean that, assuming that Ho is true, the sample proportion p-hat = 0.19 is 0.91 standard errors above the null value (0.157).

Since the null hypothesis is Ho: p = 0.64, the standardized (z) score of p-hat = 0.675 is

\(z=\dfrac{0.675-0.64}{\sqrt{\dfrac{0.64(1-0.64)}{1000}}} \approx 2.31\)

We interpret this to mean that, assuming that Ho is true, the sample proportion p-hat = 0.675 is 2.31 standard errors above the null value (0.64).

Learn by Doing: Proportions (Step 2)

Comments about the Test Statistic:

• We mentioned earlier that to some degree, the test statistic captures the essence of the test. In this case, the test statistic measures the difference between p-hat and p 0 in standard errors. This is exactly what this test is about. Get data, and look at the discrepancy between what the data estimates p to be (represented by p-hat) and what Ho claims about p (represented by p 0 ).
• You can think about this test statistic as a measure of evidence in the data against Ho. The larger the test statistic, the “further the data are from Ho” and therefore the more evidence the data provide against Ho.

Learn by Doing: Proportions (Step 2) Understanding the Test Statistic

Did I Get This?: Proportions (Step 2)

• It should now be clear why this test is commonly known as the z-test for the population proportion . The name comes from the fact that it is based on a test statistic that is a z-score.
• Recall fact 1 that we used for constructing the z-test statistic. Here is part of it again:

When we take a random sample of size n from a population with population proportion p 0 , the possible values of the sample proportion p-hat ( when certain conditions are met ) have approximately a normal distribution with a mean of p 0 … and a standard deviation of

This result provides the theoretical justification for constructing the test statistic the way we did, and therefore the assumptions under which this result holds (in bold, above) are the conditions that our data need to satisfy so that we can use this test. These two conditions are:

i. The sample has to be random.

ii. The conditions under which the sampling distribution of p-hat is normal are met. In other words:

• Here we will pause to say more about condition (i.) above, the need for a random sample. In the Probability Unit we discussed sampling plans based on probability (such as a simple random sample, cluster, or stratified sampling) that produce a non-biased sample, which can be safely used in order to make inferences about a population. We noted in the Probability Unit that, in practice, other (non-random) sampling techniques are sometimes used when random sampling is not feasible. It is important though, when these techniques are used, to be aware of the type of bias that they introduce, and thus the limitations of the conclusions that can be drawn from them. For our purpose here, we will focus on one such practice, the situation in which a sample is not really chosen randomly, but in the context of the categorical variable that is being studied, the sample is regarded as random. For example, say that you are interested in the proportion of students at a certain college who suffer from seasonal allergies. For that purpose, the students in a large engineering class could be considered as a random sample, since there is nothing about being in an engineering class that makes you more or less likely to suffer from seasonal allergies. Technically, the engineering class is a convenience sample, but it is treated as a random sample in the context of this categorical variable. On the other hand, if you are interested in the proportion of students in the college who have math anxiety, then the class of engineering students clearly could not possibly be viewed as a random sample, since engineering students probably have a much lower incidence of math anxiety than the college population overall.

Learn by Doing: Proportions (Step 2) Valid or Invalid Sampling?

Let’s check the conditions in our three examples.

i. The 400 products were chosen at random.

ii. n = 400, p 0 = 0.2 and therefore:

\(n p_{0}=400(0.2)=80 \geq 10\)

\(n\left(1-p_{0}\right)=400(1-0.2)=320 \geq 10\)

i. The 100 students were chosen at random.

ii. n = 100, p 0 = 0.157 and therefore:

\begin{gathered} n p_{0}=100(0.157)=15.7 \geq 10 \\ n\left(1-p_{0}\right)=100(1-0.157)=84.3 \geq 10 \end{gathered}

i. The 1000 adults were chosen at random.

ii. n = 1000, p 0 = 0.64 and therefore:

\begin{gathered} n p_{0}=1000(0.64)=640 \geq 10 \\ n\left(1-p_{0}\right)=1000(1-0.64)=360 \geq 10 \end{gathered}

Learn by Doing: Proportions (Step 2) Verify Conditions

Checking that our data satisfy the conditions under which the test can be reliably used is a very important part of the hypothesis testing process. Be sure to consider this for every hypothesis test you conduct in this course and certainly in practice.

The Four Steps in Hypothesis Testing

With respect to the z-test, the population proportion that we are currently discussing we have:

Step 1: Completed

Step 2: Completed

Step 3: This is what we will work on next.

Proportions (Step 3)

Video: Proportions (Step 3) (14:46)

Calculators and Tables

Step 3. Finding the P-value of the Test

So far we’ve talked about the p-value at the intuitive level: understanding what it is (or what it measures) and how we use it to draw conclusions about the statistical significance of our results. We will now go more deeply into how the p-value is calculated.

It should be mentioned that eventually we will rely on technology to calculate the p-value for us (as well as the test statistic), but in order to make intelligent use of the output, it is important to first understand the details, and only then let the computer do the calculations for us. Again, our goal is to use this simple example to give you the tools you need to understand the process entirely. Let’s start.

Recall that so far we have said that the p-value is the probability of obtaining data like those observed assuming that Ho is true. Like the test statistic, the p-value is, therefore, a measure of the evidence against Ho. In the case of the test statistic, the larger it is in magnitude (positive or negative), the further p-hat is from p 0 , the more evidence we have against Ho. In the case of the p-value , it is the opposite; the smaller it is, the more unlikely it is to get data like those observed when Ho is true, the more evidence it is against Ho . One can actually draw conclusions in hypothesis testing just using the test statistic, and as we’ll see the p-value is, in a sense, just another way of looking at the test statistic. The reason that we actually take the extra step in this course and derive the p-value from the test statistic is that even though in this case (the test about the population proportion) and some other tests, the value of the test statistic has a very clear and intuitive interpretation, there are some tests where its value is not as easy to interpret. On the other hand, the p-value keeps its intuitive appeal across all statistical tests.

How is the p-value calculated?

Intuitively, the p-value is the probability of observing data like those observed assuming that Ho is true. Let’s be a bit more formal:

• Since this is a probability question about the data , it makes sense that the calculation will involve the data summary, the test statistic.
• What do we mean by “like” those observed? By “like” we mean “as extreme or even more extreme.”

Putting it all together, we get that in general:

The p-value is the probability of observing a test statistic as extreme as that observed (or even more extreme) assuming that the null hypothesis is true.

By “extreme” we mean extreme in the direction(s) of the alternative hypothesis.

Specifically , for the z-test for the population proportion:

• If the alternative hypothesis is Ha: p < p 0 (less than) , then “extreme” means small or less than , and the p-value is: The probability of observing a test statistic as small as that observed or smaller if the null hypothesis is true.
• If the alternative hypothesis is Ha: p > p 0 (greater than) , then “extreme” means large or greater than , and the p-value is: The probability of observing a test statistic as large as that observed or larger if the null hypothesis is true.
• If the alternative is Ha: p ≠ p 0 (different from) , then “extreme” means extreme in either direction either small or large (i.e., large in magnitude) or just different from , and the p-value therefore is: The probability of observing a test statistic as large in magnitude as that observed or larger if the null hypothesis is true.(Examples: If z = -2.5: p-value = probability of observing a test statistic as small as -2.5 or smaller or as large as 2.5 or larger. If z = 1.5: p-value = probability of observing a test statistic as large as 1.5 or larger, or as small as -1.5 or smaller.)

OK, hopefully that makes (some) sense. But how do we actually calculate it?

Recall the important comment from our discussion about our test statistic,

which said that when the null hypothesis is true (i.e., when p = p 0 ), the possible values of our test statistic follow a standard normal (N(0,1), denoted by Z) distribution. Therefore, the p-value calculations (which assume that Ho is true) are simply standard normal distribution calculations for the 3 possible alternative hypotheses.

Alternative Hypothesis is “Less Than”

The probability of observing a test statistic as small as that observed or smaller , assuming that the values of the test statistic follow a standard normal distribution. We will now represent this probability in symbols and also using the normal distribution.

Looking at the shaded region, you can see why this is often referred to as a left-tailed test. We shaded to the left of the test statistic, since less than is to the left.

Alternative Hypothesis is “Greater Than”

The probability of observing a test statistic as large as that observed or larger , assuming that the values of the test statistic follow a standard normal distribution. Again, we will represent this probability in symbols and using the normal distribution

Looking at the shaded region, you can see why this is often referred to as a right-tailed test. We shaded to the right of the test statistic, since greater than is to the right.

Alternative Hypothesis is “Not Equal To”

The probability of observing a test statistic which is as large in magnitude as that observed or larger, assuming that the values of the test statistic follow a standard normal distribution.

This is often referred to as a two-tailed test, since we shaded in both directions.

Next, we will apply this to our three examples. But first, work through the following activities, which should help your understanding.

Learn by Doing: Proportions (Step 3)

Did I Get This?: Proportions (Step 3)

The p-value in this case is:

• The probability of observing a test statistic as small as -2 or smaller, assuming that Ho is true.

OR (recalling what the test statistic actually means in this case),

• The probability of observing a sample proportion that is 2 standard deviations or more below the null value (p 0 = 0.20), assuming that p 0 is the true population proportion.

OR, more specifically,

• The probability of observing a sample proportion of 0.16 or lower in a random sample of size 400, when the true population proportion is p 0 =0.20

In either case, the p-value is found as shown in the following figure:

To find P(Z ≤ -2) we can either use the calculator or table we learned to use in the probability unit for normal random variables. Eventually, after we understand the details, we will use software to run the test for us and the output will give us all the information we need. The p-value that the statistical software provides for this specific example is 0.023. The p-value tells us that it is pretty unlikely (probability of 0.023) to get data like those observed (test statistic of -2 or less) assuming that Ho is true.

• The probability of observing a test statistic as large as 0.91 or larger, assuming that Ho is true.
• The probability of observing a sample proportion that is 0.91 standard deviations or more above the null value (p 0 = 0.157), assuming that p 0 is the true population proportion.
• The probability of observing a sample proportion of 0.19 or higher in a random sample of size 100, when the true population proportion is p 0 =0.157

Again, at this point we can either use the calculator or table to find that the p-value is 0.182, this is P(Z ≥ 0.91).

The p-value tells us that it is not very surprising (probability of 0.182) to get data like those observed (which yield a test statistic of 0.91 or higher) assuming that the null hypothesis is true.

• The probability of observing a test statistic as large as 2.31 (or larger) or as small as -2.31 (or smaller), assuming that Ho is true.
• The probability of observing a sample proportion that is 2.31 standard deviations or more away from the null value (p 0 = 0.64), assuming that p 0 is the true population proportion.
• The probability of observing a sample proportion as different as 0.675 is from 0.64, or even more different (i.e. as high as 0.675 or higher or as low as 0.605 or lower) in a random sample of size 1,000, when the true population proportion is p 0 = 0.64

Again, at this point we can either use the calculator or table to find that the p-value is 0.021, this is P(Z ≤ -2.31) + P(Z ≥ 2.31) = 2*P(Z ≥ |2.31|)

The p-value tells us that it is pretty unlikely (probability of 0.021) to get data like those observed (test statistic as high as 2.31 or higher or as low as -2.31 or lower) assuming that Ho is true.

• We’ve just seen that finding p-values involves probability calculations about the value of the test statistic assuming that Ho is true. In this case, when Ho is true, the values of the test statistic follow a standard normal distribution (i.e., the sampling distribution of the test statistic when the null hypothesis is true is N(0,1)). Therefore, p-values correspond to areas (probabilities) under the standard normal curve.

Similarly, in any test , p-values are found using the sampling distribution of the test statistic when the null hypothesis is true (also known as the “null distribution” of the test statistic). In this case, it was relatively easy to argue that the null distribution of our test statistic is N(0,1). As we’ll see, in other tests, other distributions come up (like the t-distribution and the F-distribution), which we will just mention briefly, and rely heavily on the output of our statistical package for obtaining the p-values.

We’ve just completed our discussion about the p-value, and how it is calculated both in general and more specifically for the z-test for the population proportion. Let’s go back to the four-step process of hypothesis testing and see what we’ve covered and what still needs to be discussed.

With respect to the z-test the population proportion:

Step 3: Completed

Step 4. This is what we will work on next.

Learn by Doing: Proportions (Step 3) Understanding P-values

Proportions (Step 4 & Summary)

Video: Proportions (Step 4 & Summary) (4:30)

Step 4. Drawing Conclusions Based on the P-Value

This last part of the four-step process of hypothesis testing is the same across all statistical tests, and actually, we’ve already said basically everything there is to say about it, but it can’t hurt to say it again.

The p-value is a measure of how much evidence the data present against Ho. The smaller the p-value, the more evidence the data present against Ho.

We already mentioned that what determines what constitutes enough evidence against Ho is the significance level (α, alpha), a cutoff point below which the p-value is considered small enough to reject Ho in favor of Ha. The most commonly used significance level is 0.05.

• Conclusion: There IS enough evidence that Ha is True
• Conclusion: There IS NOT enough evidence that Ha is True

Where instead of Ha is True , we write what this means in the words of the problem, in other words, in the context of the current scenario.

It is important to mention again that this step has essentially two sub-steps:

(i) Based on the p-value, determine whether or not the results are statistically significant (i.e., the data present enough evidence to reject Ho).

(ii) State your conclusions in the context of the problem.

Note: We always still must consider whether the results have any practical significance, particularly if they are statistically significant as a statistically significant result which has not practical use is essentially meaningless!

Let’s go back to our three examples and draw conclusions.

We found that the p-value for this test was 0.023.

Since 0.023 is small (in particular, 0.023 < 0.05), the data provide enough evidence to reject Ho.

Conclusion:

• There IS enough evidence that the proportion of defective products is less than 20% after the repair .

The following figure is the complete story of this example, and includes all the steps we went through, starting from stating the hypotheses and ending with our conclusions:

We found that the p-value for this test was 0.182.

Since .182 is not small (in particular, 0.182 > 0.05), the data do not provide enough evidence to reject Ho.

• There IS NOT enough evidence that the proportion of students at the college who use marijuana is higher than the national figure.

Here is the complete story of this example:

Learn by Doing: Learn by Doing – Proportions (Step 4)

We found that the p-value for this test was 0.021.

Since 0.021 is small (in particular, 0.021 < 0.05), the data provide enough evidence to reject Ho

• There IS enough evidence that the proportion of adults who support the death penalty for convicted murderers has changed since 2003.

Did I Get This?: Proportions (Step 4)

Many Students Wonder: Hypothesis Testing for the Population Proportion

Many students wonder why 5% is often selected as the significance level in hypothesis testing, and why 1% is the next most typical level. This is largely due to just convenience and tradition.

When Ronald Fisher (one of the founders of modern statistics) published one of his tables, he used a mathematically convenient scale that included 5% and 1%. Later, these same 5% and 1% levels were used by other people, in part just because Fisher was so highly esteemed. But mostly these are arbitrary levels.

The idea of selecting some sort of relatively small cutoff was historically important in the development of statistics; but it’s important to remember that there is really a continuous range of increasing confidence towards the alternative hypothesis, not a single all-or-nothing value. There isn’t much meaningful difference, for instance, between a p-value of .049 or .051, and it would be foolish to declare one case definitely a “real” effect and to declare the other case definitely a “random” effect. In either case, the study results were roughly 5% likely by chance if there’s no actual effect.

Whether such a p-value is sufficient for us to reject a particular null hypothesis ultimately depends on the risk of making the wrong decision, and the extent to which the hypothesized effect might contradict our prior experience or previous studies.

Let’s Summarize!!

We have now completed going through the four steps of hypothesis testing, and in particular we learned how they are applied to the z-test for the population proportion. Here is a brief summary:

Step 1: State the hypotheses

State the null hypothesis:

State the alternative hypothesis:

where the choice of the appropriate alternative (out of the three) is usually quite clear from the context of the problem. If you feel it is not clear, it is most likely a two-sided problem. Students are usually good at recognizing the “more than” and “less than” terminology but differences can sometimes be more difficult to spot, sometimes this is because you have preconceived ideas of how you think it should be! Use only the information given in the problem.

Step 2: Obtain data, check conditions, and summarize data

Obtain data from a sample and:

(i) Check whether the data satisfy the conditions which allow you to use this test.

random sample (or at least a sample that can be considered random in context)

the conditions under which the sampling distribution of p-hat is normal are met

(ii) Calculate the sample proportion p-hat, and summarize the data using the test statistic:

( Recall: This standardized test statistic represents how many standard deviations above or below p 0 our sample proportion p-hat is.)

Step 3: Find the p-value of the test by using the test statistic as follows

IMPORTANT FACT: In all future tests, we will rely on software to obtain the p-value.

When the alternative hypothesis is “less than” the probability of observing a test statistic as small as that observed or smaller , assuming that the values of the test statistic follow a standard normal distribution. We will now represent this probability in symbols and also using the normal distribution.

When the alternative hypothesis is “greater than” the probability of observing a test statistic as large as that observed or larger , assuming that the values of the test statistic follow a standard normal distribution. Again, we will represent this probability in symbols and using the normal distribution

When the alternative hypothesis is “not equal to” the probability of observing a test statistic which is as large in magnitude as that observed or larger, assuming that the values of the test statistic follow a standard normal distribution.

Step 4: Conclusion

Reach a conclusion first regarding the statistical significance of the results, and then determine what it means in the context of the problem.

If p-value ≤ 0.05 then WE REJECT Ho Conclusion: There IS enough evidence that Ha is True

If p-value > 0.05 then WE FAIL TO REJECT Ho Conclusion: There IS NOT enough evidence that Ha is True

Recall that: If the p-value is small (in particular, smaller than the significance level, which is usually 0.05), the results are statistically significant (in the sense that there is a statistically significant difference between what was observed in the sample and what was claimed in Ho), and so we reject Ho.

If the p-value is not small, we do not have enough statistical evidence to reject Ho, and so we continue to believe that Ho may be true. ( Remember: In hypothesis testing we never “accept” Ho ).

Finally, in practice, we should always consider the practical significance of the results as well as the statistical significance.

Learn by Doing: Z-Test for a Population Proportion

What’s next?

Before we move on to the next test, we are going to use the z-test for proportions to bring up and illustrate a few more very important issues regarding hypothesis testing. This might also be a good time to review the concepts of Type I error, Type II error, and Power before continuing on.

More about Hypothesis Testing

CO-1: Describe the roles biostatistics serves in the discipline of public health.

LO 1.11: Recognize the distinction between statistical significance and practical significance.

LO 6.30: Use a confidence interval to determine the correct conclusion to the associated two-sided hypothesis test.

Video: More about Hypothesis Testing (18:25)

The issues regarding hypothesis testing that we will discuss are:

• The effect of sample size on hypothesis testing.
• Statistical significance vs. practical importance.
• Hypothesis testing and confidence intervals—how are they related?

Let’s begin.

1. The Effect of Sample Size on Hypothesis Testing

We have already seen the effect that the sample size has on inference, when we discussed point and interval estimation for the population mean (μ, mu) and population proportion (p). Intuitively …

Larger sample sizes give us more information to pin down the true nature of the population. We can therefore expect the sample mean and sample proportion obtained from a larger sample to be closer to the population mean and proportion, respectively. As a result, for the same level of confidence, we can report a smaller margin of error, and get a narrower confidence interval. What we’ve seen, then, is that larger sample size gives a boost to how much we trust our sample results.

In hypothesis testing, larger sample sizes have a similar effect. We have also discussed that the power of our test increases when the sample size increases, all else remaining the same. This means, we have a better chance to detect the difference between the true value and the null value for larger samples.

The following two examples will illustrate that a larger sample size provides more convincing evidence (the test has greater power), and how the evidence manifests itself in hypothesis testing. Let’s go back to our example 2 (marijuana use at a certain liberal arts college).

We do not have enough evidence to conclude that the proportion of students at the college who use marijuana is higher than the national figure.

Now, let’s increase the sample size.

There are rumors that students in a certain liberal arts college are more inclined to use drugs than U.S. college students in general. Suppose that in a simple random sample of 400 students from the college, 76 admitted to marijuana use . Do the data provide enough evidence to conclude that the proportion of marijuana users among the students in the college (p) is higher than the national proportion, which is 0.157? (Reported by the Harvard School of Public Health).

Our results here are statistically significant . In other words, in example 2* the data provide enough evidence to reject Ho.

• Conclusion: There is enough evidence that the proportion of marijuana users at the college is higher than among all U.S. students.

What do we learn from this?

We see that sample results that are based on a larger sample carry more weight (have greater power).

In example 2, we saw that a sample proportion of 0.19 based on a sample of size of 100 was not enough evidence that the proportion of marijuana users in the college is higher than 0.157. Recall, from our general overview of hypothesis testing, that this conclusion (not having enough evidence to reject the null hypothesis) doesn’t mean the null hypothesis is necessarily true (so, we never “accept” the null); it only means that the particular study didn’t yield sufficient evidence to reject the null. It might be that the sample size was simply too small to detect a statistically significant difference.

However, in example 2*, we saw that when the sample proportion of 0.19 is obtained from a sample of size 400, it carries much more weight, and in particular, provides enough evidence that the proportion of marijuana users in the college is higher than 0.157 (the national figure). In this case, the sample size of 400 was large enough to detect a statistically significant difference.

The following activity will allow you to practice the ideas and terminology used in hypothesis testing when a result is not statistically significant.

Learn by Doing: Interpreting Non-significant Results

2. Statistical significance vs. practical importance.

Now, we will address the issue of statistical significance versus practical importance (which also involves issues of sample size).

The following activity will let you explore the effect of the sample size on the statistical significance of the results yourself, and more importantly will discuss issue 2: Statistical significance vs. practical importance.

Important Fact: In general, with a sufficiently large sample size you can make any result that has very little practical importance statistically significant! A large sample size alone does NOT make a “good” study!!

This suggests that when interpreting the results of a test, you should always think not only about the statistical significance of the results but also about their practical importance.

Learn by Doing: Statistical vs. Practical Significance

3. Hypothesis Testing and Confidence Intervals

The last topic we want to discuss is the relationship between hypothesis testing and confidence intervals. Even though the flavor of these two forms of inference is different (confidence intervals estimate a parameter, and hypothesis testing assesses the evidence in the data against one claim and in favor of another), there is a strong link between them.

We will explain this link (using the z-test and confidence interval for the population proportion), and then explain how confidence intervals can be used after a test has been carried out.

Recall that a confidence interval gives us a set of plausible values for the unknown population parameter. We may therefore examine a confidence interval to informally decide if a proposed value of population proportion seems plausible.

For example, if a 95% confidence interval for p, the proportion of all U.S. adults already familiar with Viagra in May 1998, was (0.61, 0.67), then it seems clear that we should be able to reject a claim that only 50% of all U.S. adults were familiar with the drug, since based on the confidence interval, 0.50 is not one of the plausible values for p.

In fact, the information provided by a confidence interval can be formally related to the information provided by a hypothesis test. ( Comment: The relationship is more straightforward for two-sided alternatives, and so we will not present results for the one-sided cases.)

Suppose we want to carry out the two-sided test:

• Ha: p ≠ p 0

using a significance level of 0.05.

An alternative way to perform this test is to find a 95% confidence interval for p and check:

• If p 0 falls outside the confidence interval, reject Ho.
• If p 0 falls inside the confidence interval, do not reject Ho.

In other words,

• If p 0 is not one of the plausible values for p, we reject Ho.
• If p 0 is a plausible value for p, we cannot reject Ho.

( Comment: Similarly, the results of a test using a significance level of 0.01 can be related to the 99% confidence interval.)

Let’s look at an example:

Recall example 3, where we wanted to know whether the proportion of U.S. adults who support the death penalty for convicted murderers has changed since 2003, when it was 0.64.

We are testing:

and as the figure reminds us, we took a sample of 1,000 U.S. adults, and the data told us that 675 supported the death penalty for convicted murderers (p-hat = 0.675).

A 95% confidence interval for p, the proportion of all U.S. adults who support the death penalty, is:

\(0.675 \pm 1.96 \sqrt{\dfrac{0.675(1-0.675)}{1000}} \approx 0.675 \pm 0.029=(0.646,0.704)\)

Since the 95% confidence interval for p does not include 0.64 as a plausible value for p, we can reject Ho and conclude (as we did before) that there is enough evidence that the proportion of U.S. adults who support the death penalty for convicted murderers has changed since 2003.

You and your roommate are arguing about whose turn it is to clean the apartment. Your roommate suggests that you settle this by tossing a coin and takes one out of a locked box he has on the shelf. Suspecting that the coin might not be fair, you decide to test it first. You toss the coin 80 times, thinking to yourself that if, indeed, the coin is fair, you should get around 40 heads. Instead you get 48 heads. You are puzzled. You are not sure whether getting 48 heads out of 80 is enough evidence to conclude that the coin is unbalanced, or whether this a result that could have happened just by chance when the coin is fair.

Statistics can help you answer this question.

Let p be the true proportion (probability) of heads. We want to test whether the coin is fair or not.

• Ho: p = 0.5 (the coin is fair).
• Ha: p ≠ 0.5 (the coin is not fair).

The data we have are that out of n = 80 tosses, we got 48 heads, or that the sample proportion of heads is p-hat = 48/80 = 0.6.

A 95% confidence interval for p, the true proportion of heads for this coin, is:

\(0.6 \pm 1.96 \sqrt{\dfrac{0.6(1-0.6)}{80}} \approx 0.6 \pm 0.11=(0.49,0.71)\)

Since in this case 0.5 is one of the plausible values for p, we cannot reject Ho. In other words, the data do not provide enough evidence to conclude that the coin is not fair.

The context of the last example is a good opportunity to bring up an important point that was discussed earlier.

Even though we use 0.05 as a cutoff to guide our decision about whether the results are statistically significant, we should not treat it as inviolable and we should always add our own judgment. Let’s look at the last example again.

It turns out that the p-value of this test is 0.0734. In other words, it is maybe not extremely unlikely, but it is quite unlikely (probability of 0.0734) that when you toss a fair coin 80 times you’ll get a sample proportion of heads of 48/80 = 0.6 (or even more extreme). It is true that using the 0.05 significance level (cutoff), 0.0734 is not considered small enough to conclude that the coin is not fair. However, if you really don’t want to clean the apartment, the p-value might be small enough for you to ask your roommate to use a different coin, or to provide one yourself!

Did I Get This?: Connection between Confidence Intervals and Hypothesis Tests

Did I Get This?: Hypothesis Tests for Proportions (Extra Practice)

Here is our final point on this subject:

When the data provide enough evidence to reject Ho, we can conclude (depending on the alternative hypothesis) that the population proportion is either less than, greater than, or not equal to the null value p 0 . However, we do not get a more informative statement about its actual value. It might be of interest, then, to follow the test with a 95% confidence interval that will give us more insight into the actual value of p.

In our example 3,

we concluded that the proportion of U.S. adults who support the death penalty for convicted murderers has changed since 2003, when it was 0.64. It is probably of interest not only to know that the proportion has changed, but also to estimate what it has changed to. We’ve calculated the 95% confidence interval for p on the previous page and found that it is (0.646, 0.704).

We can combine our conclusions from the test and the confidence interval and say:

Data provide evidence that the proportion of U.S. adults who support the death penalty for convicted murderers has changed since 2003, and we are 95% confident that it is now between 0.646 and 0.704. (i.e. between 64.6% and 70.4%).

Let’s look at our example 1 to see how a confidence interval following a test might be insightful in a different way.

Here is a summary of example 1:

We conclude that as a result of the repair, the proportion of defective products has been reduced to below 0.20 (which was the proportion prior to the repair). It is probably of great interest to the company not only to know that the proportion of defective has been reduced, but also estimate what it has been reduced to, to get a better sense of how effective the repair was. A 95% confidence interval for p in this case is:

\(0.16 \pm 1.96 \sqrt{\dfrac{0.16(1-0.16)}{400}} \approx 0.16 \pm 0.036=(0.124,0.196)\)

We can therefore say that the data provide evidence that the proportion of defective products has been reduced, and we are 95% confident that it has been reduced to somewhere between 12.4% and 19.6%. This is very useful information, since it tells us that even though the results were significant (i.e., the repair reduced the number of defective products), the repair might not have been effective enough, if it managed to reduce the number of defective products only to the range provided by the confidence interval. This, of course, ties back in to the idea of statistical significance vs. practical importance that we discussed earlier. Even though the results are statistically significant (Ho was rejected), practically speaking, the repair might still be considered ineffective.

Learn by Doing: Hypothesis Tests and Confidence Intervals

Even though this portion of the current section is about the z-test for population proportion, it is loaded with very important ideas that apply to hypothesis testing in general. We’ve already summarized the details that are specific to the z-test for proportions, so the purpose of this summary is to highlight the general ideas.

The process of hypothesis testing has four steps :

I. Stating the null and alternative hypotheses (Ho and Ha).

II. Obtaining a random sample (or at least one that can be considered random) and collecting data. Using the data:

Check that the conditions under which the test can be reliably used are met.

Summarize the data using a test statistic.

• The test statistic is a measure of the evidence in the data against Ho. The larger the test statistic is in magnitude, the more evidence the data present against Ho.

III. Finding the p-value of the test. The p-value is the probability of getting data like those observed (or even more extreme) assuming that the null hypothesis is true, and is calculated using the null distribution of the test statistic. The p-value is a measure of the evidence against Ho. The smaller the p-value, the more evidence the data present against Ho.

IV. Making conclusions.

Conclusions about the statistical significance of the results:

If the p-value is small, the data present enough evidence to reject Ho (and accept Ha).

If the p-value is not small, the data do not provide enough evidence to reject Ho.

To help guide our decision, we use the significance level as a cutoff for what is considered a small p-value. The significance cutoff is usually set at 0.05.

Conclusions should then be provided in the context of the problem.

Additional Important Ideas about Hypothesis Testing

• Results that are based on a larger sample carry more weight, and therefore as the sample size increases, results become more statistically significant.
• Even a very small and practically unimportant effect becomes statistically significant with a large enough sample size. The distinction between statistical significance and practical importance should therefore always be considered.
• Confidence intervals can be used in order to carry out two-sided tests (95% confidence for the 0.05 significance level). If the null value is not included in the confidence interval (i.e., is not one of the plausible values for the parameter), we have enough evidence to reject Ho. Otherwise, we cannot reject Ho.
• If the results are statistically significant, it might be of interest to follow up the tests with a confidence interval in order to get insight into the actual value of the parameter of interest.
• It is important to be aware that there are two types of errors in hypothesis testing ( Type I and Type II ) and that the power of a statistical test is an important measure of how likely we are to be able to detect a difference of interest to us in a particular problem.

Means (All Steps)

NOTE: Beginning on this page, the Learn By Doing and Did I Get This activities are presented as interactive PDF files. The interactivity may not work on mobile devices or with certain PDF viewers. Use an official ADOBE product such as ADOBE READER .

If you have any issues with the Learn By Doing or Did I Get This interactive PDF files, you can view all of the questions and answers presented on this page in this document:

• QUESTION/Answer (SPOILER ALERT!)

Tests About μ (mu) When σ (sigma) is Unknown – The t-test for a Population Mean

The t-distribution.

Video: Means (All Steps) (13:11)

So far we have talked about the logic behind hypothesis testing and then illustrated how this process proceeds in practice, using the z-test for the population proportion (p).

We are now moving on to discuss testing for the population mean (μ, mu), which is the parameter of interest when the variable of interest is quantitative.

A few comments about the structure of this section:

• The basic groundwork for carrying out hypothesis tests has already been laid in our general discussion and in our presentation of tests about proportions.

Therefore we can easily modify the four steps to carry out tests about means instead, without going into all of the details again.

We will use this approach for all future tests so be sure to go back to the discussion in general and for proportions to review the concepts in more detail.

• In our discussion about confidence intervals for the population mean, we made the distinction between whether the population standard deviation, σ (sigma) was known or if we needed to estimate this value using the sample standard deviation, s .

In this section, we will only discuss the second case as in most realistic settings we do not know the population standard deviation .

In this case we need to use the t- distribution instead of the standard normal distribution for the probability aspects of confidence intervals (choosing table values) and hypothesis tests (finding p-values).

• Although we will discuss some theoretical or conceptual details for some of the analyses we will learn, from this point on we will rely on software to conduct tests and calculate confidence intervals for us , while we focus on understanding which methods are used for which situations and what the results say in context.

If you are interested in more information about the z-test, where we assume the population standard deviation σ (sigma) is known, you can review the Carnegie Mellon Open Learning Statistics Course (you will need to click “ENTER COURSE”).

Like any other tests, the t- test for the population mean follows the four-step process:

• STEP 1: Stating the hypotheses H o and H a .
• STEP 2: Collecting relevant data, checking that the data satisfy the conditions which allow us to use this test, and summarizing the data using a test statistic.
• STEP 3: Finding the p-value of the test, the probability of obtaining data as extreme as those collected (or even more extreme, in the direction of the alternative hypothesis), assuming that the null hypothesis is true. In other words, how likely is it that the only reason for getting data like those observed is sampling variability (and not because H o is not true)?
• STEP 4: Drawing conclusions, assessing the statistical significance of the results based on the p-value, and stating our conclusions in context. (Do we or don’t we have evidence to reject H o and accept H a ?)
• Note: In practice, we should also always consider the practical significance of the results as well as the statistical significance.

We will now go through the four steps specifically for the t- test for the population mean and apply them to our two examples.

Only in a few cases is it reasonable to assume that the population standard deviation, σ (sigma), is known and so we will not cover hypothesis tests in this case. We discussed both cases for confidence intervals so that we could still calculate some confidence intervals by hand.

For this and all future tests we will rely on software to obtain our summary statistics, test statistics, and p-values for us.

The case where σ (sigma) is unknown is much more common in practice. What can we use to replace σ (sigma)? If you don’t know the population standard deviation, the best you can do is find the sample standard deviation, s, and use it instead of σ (sigma). (Note that this is exactly what we did when we discussed confidence intervals).

Is that it? Can we just use s instead of σ (sigma), and the rest is the same as the previous case? Unfortunately, it’s not that simple, but not very complicated either.

Here, when we use the sample standard deviation, s, as our estimate of σ (sigma) we can no longer use a normal distribution to find the cutoff for confidence intervals or the p-values for hypothesis tests.

Instead we must use the t- distribution (with n-1 degrees of freedom) to obtain the p-value for this test.

We discussed this issue for confidence intervals. We will talk more about the t- distribution after we discuss the details of this test for those who are interested in learning more.

It isn’t really necessary for us to understand this distribution but it is important that we use the correct distributions in practice via our software.

We will wait until UNIT 4B to look at how to accomplish this test in the software. For now focus on understanding the process and drawing the correct conclusions from the p-values given.

Now let’s go through the four steps in conducting the t- test for the population mean.

The null and alternative hypotheses for the t- test for the population mean (μ, mu) have exactly the same structure as the hypotheses for z-test for the population proportion (p):

The null hypothesis has the form:

• Ho: μ = μ 0 (mu = mu_zero)

(where μ 0 (mu_zero) is often called the null value)

• Ha: μ < μ 0 (mu < mu_zero) (one-sided)
• Ha: μ > μ 0 (mu > mu_zero) (one-sided)
• Ha: μ ≠ μ 0 (mu ≠ mu_zero) (two-sided)

where the choice of the appropriate alternative (out of the three) is usually quite clear from the context of the problem.

If you feel it is not clear, it is most likely a two-sided problem. Students are usually good at recognizing the “more than” and “less than” terminology but differences can sometimes be more difficult to spot, sometimes this is because you have preconceived ideas of how you think it should be! You also cannot use the information from the sample to help you determine the hypothesis. We would not know our data when we originally asked the question.

Now try it yourself. Here are a few exercises on stating the hypotheses for tests for a population mean.

Learn by Doing: State the Hypotheses for a test for a population mean

Here are a few more activities for practice.

Did I Get This?: State the Hypotheses for a test for a population mean

When setting up hypotheses, be sure to use only the information in the research question. We cannot use our sample data to help us set up our hypotheses.

For this test, it is still important to correctly choose the alternative hypothesis as “less than”, “greater than”, or “different” although generally in practice two-sample tests are used.

Obtain data from a sample:

• In this step we would obtain data from a sample. This is not something we do much of in courses but it is done very often in practice!

Check the conditions:

• Then we check the conditions under which this test (the t- test for one population mean) can be safely carried out – which are:
• The sample is random (or at least can be considered random in context).
• We are in one of the three situations marked with a green check mark in the following table (which ensure that x-bar is at least approximately normal and the test statistic using the sample standard deviation, s, is therefore a t- distribution with n-1 degrees of freedom – proving this is beyond the scope of this course):
• For large samples, we don’t need to check for normality in the population . We can rely on the sample size as the basis for the validity of using this test.
• For small samples , we need to have data from a normal population in order for the p-values and confidence intervals to be valid.

In practice, for small samples, it can be very difficult to determine if the population is normal. Here is a simulation to give you a better understanding of the difficulties.

Video: Simulations – Are Samples from a Normal Population? (4:58)

Now try it yourself with a few activities.

Learn by Doing: Checking Conditions for Hypothesis Testing for the Population Mean

• It is always a good idea to look at the data and get a sense of their pattern regardless of whether you actually need to do it in order to assess whether the conditions are met.
• This idea of looking at the data is relevant to all tests in general. In the next module—inference for relationships—conducting exploratory data analysis before inference will be an integral part of the process.

Here are a few more problems for extra practice.

Did I Get This?: Checking Conditions for Hypothesis Testing for the Population Mean

When setting up hypotheses, be sure to use only the information in the res

Calculate Test Statistic

Assuming that the conditions are met, we calculate the sample mean x-bar and the sample standard deviation, s (which estimates σ (sigma)), and summarize the data with a test statistic.

The test statistic for the t -test for the population mean is:

\(t=\dfrac{\bar{x} - \mu_0}{s/ \sqrt{n}}\)

Recall that such a standardized test statistic represents how many standard deviations above or below μ 0 (mu_zero) our sample mean x-bar is.

Therefore our test statistic is a measure of how different our data are from what is claimed in the null hypothesis. This is an idea that we mentioned in the previous test as well.

Again we will rely on the p-value to determine how unusual our data would be if the null hypothesis is true.

As we mentioned, the test statistic in the t -test for a population mean does not follow a standard normal distribution. Rather, it follows another bell-shaped distribution called the t- distribution.

We will present the details of this distribution at the end for those interested but for now we will work on the process of the test.

Here are a few important facts.

• In statistical language we say that the null distribution of our test statistic is the t- distribution with (n-1) degrees of freedom. In other words, when Ho is true (i.e., when μ = μ 0 (mu = mu_zero)), our test statistic has a t- distribution with (n-1) d.f., and this is the distribution under which we find p-values.
• For a large sample size (n), the null distribution of the test statistic is approximately Z, so whether we use t (n – 1) or Z to calculate the p-values does not make a big difference. However, software will use the t -distribution regardless of the sample size and so will we.

Although we will not calculate p-values by hand for this test, we can still easily calculate the test statistic.

Try it yourself:

Learn by Doing: Calculate the Test Statistic for a Test for a Population Mean

From this point in this course and certainly in practice we will allow the software to calculate our test statistics and we will use the p-values provided to draw our conclusions.

We will use software to obtain the p-value for this (and all future) tests but here are the images illustrating how the p-value is calculated in each of the three cases corresponding to the three choices for our alternative hypothesis.

Note that due to the symmetry of the t distribution, for a given value of the test statistic t, the p-value for the two-sided test is twice as large as the p-value of either of the one-sided tests. The same thing happens when p-values are calculated under the t distribution as when they are calculated under the Z distribution.

We will show some examples of p-values obtained from software in our examples. For now let’s continue our summary of the steps.

As usual, based on the p-value (and some significance level of choice) we assess the statistical significance of results, and draw our conclusions in context.

To review what we have said before:

If p-value ≤ 0.05 then WE REJECT Ho

If p-value > 0.05 then WE FAIL TO REJECT Ho

This step has essentially two sub-steps:

We are now ready to look at two examples.

A certain prescription medicine is supposed to contain an average of 250 parts per million (ppm) of a certain chemical. If the concentration is higher than this, the drug may cause harmful side effects; if it is lower, the drug may be ineffective.

The manufacturer runs a check to see if the mean concentration in a large shipment conforms to the target level of 250 ppm or not.

A simple random sample of 100 portions is tested, and the sample mean concentration is found to be 247 ppm with a sample standard deviation of 12 ppm.

Here is a figure that represents this example:

1. The hypotheses being tested are:

• Ha: μ ≠ μ 0 (mu ≠ mu_zero)
• Where μ = population mean part per million of the chemical in the entire shipment

2. The conditions that allow us to use the t-test are met since:

• The sample is random
• The sample size is large enough for the Central Limit Theorem to apply and ensure the normality of x-bar. We do not need normality of the population in order to be able to conduct this test for the population mean. We are in the 2 nd column in the table below.
• The test statistic is:

\(t=\dfrac{\bar{x}-\mu_{0}}{s / \sqrt{n}}=\dfrac{247-250}{12 / \sqrt{100}}=-2.5\)

• The data (represented by the sample mean) are 2.5 standard errors below the null value.

3. Finding the p-value.

• To find the p-value we use statistical software, and we calculate a p-value of 0.014.

4. Conclusions:

• The p-value is small (.014) indicating that at the 5% significance level, the results are significant.
• We reject the null hypothesis.
• There is enough evidence to conclude that the mean concentration in entire shipment is not the required 250 ppm.
• It is difficult to comment on the practical significance of this result without more understanding of the practical considerations of this problem.

Here is a summary:

• The 95% confidence interval for μ (mu) can be used here in the same way as for proportions to conduct the two-sided test (checking whether the null value falls inside or outside the confidence interval) or following a t- test where Ho was rejected to get insight into the value of μ (mu).
• We find the 95% confidence interval to be (244.619, 249.381) . Since 250 is not in the interval we know we would reject our null hypothesis that μ (mu) = 250. The confidence interval gives additional information. By accounting for estimation error, it estimates that the population mean is likely to be between 244.62 and 249.38. This is lower than the target concentration and that information might help determine the seriousness and appropriate course of action in this situation.

In most situations in practice we use TWO-SIDED HYPOTHESIS TESTS, followed by confidence intervals to gain more insight.

For completeness in covering one sample t-tests for a population mean, we still cover all three possible alternative hypotheses here HOWEVER, this will be the last test for which we will do so.

A research study measured the pulse rates of 57 college men and found a mean pulse rate of 70 beats per minute with a standard deviation of 9.85 beats per minute.

Researchers want to know if the mean pulse rate for all college men is different from the current standard of 72 beats per minute.

• The hypotheses being tested are:
• Ho: μ = 72
• Ha: μ ≠ 72
• Where μ = population mean heart rate among college men
• The conditions that allow us to use the t- test are met since:
• The sample is random.
• The sample size is large (n = 57) so we do not need normality of the population in order to be able to conduct this test for the population mean. We are in the 2 nd column in the table below.

\(t=\dfrac{\bar{x}-\mu}{s / \sqrt{n}}=\dfrac{70-72}{9.85 / \sqrt{57}}=-1.53\)

• The data (represented by the sample mean) are 1.53 estimated standard errors below the null value.
• Recall that in general the p-value is calculated under the null distribution of the test statistic, which, in the t- test case, is t (n-1). In our case, in which n = 57, the p-value is calculated under the t (56) distribution. Using statistical software, we find that the p-value is 0.132 .
• Here is how we calculated the p-value. http://homepage.stat.uiowa.edu/~mbognar/applets/t.html .

4. Making conclusions.

• The p-value (0.132) is not small, indicating that the results are not significant.
• We fail to reject the null hypothesis.
• There is not enough evidence to conclude that the mean pulse rate for all college men is different from the current standard of 72 beats per minute.
• The results from this sample do not appear to have any practical significance either with a mean pulse rate of 70, this is very similar to the hypothesized value, relative to the variation expected in pulse rates.

Now try a few yourself.

Learn by Doing: Hypothesis Testing for the Population Mean

From this point in this course and certainly in practice we will allow the software to calculate our test statistic and p-value and we will use the p-values provided to draw our conclusions.

That concludes our discussion of hypothesis tests in Unit 4A.

In the next unit we will continue to use both confidence intervals and hypothesis test to investigate the relationship between two variables in the cases we covered in Unit 1 on exploratory data analysis – we will look at Case CQ, Case CC, and Case QQ.

Before moving on, we will discuss the details about the t- distribution as a general object.

We have seen that variables can be visually modeled by many different sorts of shapes, and we call these shapes distributions. Several distributions arise so frequently that they have been given special names, and they have been studied mathematically.

So far in the course, the only one we’ve named, for continuous quantitative variables, is the normal distribution, but there are others. One of them is called the t- distribution.

The t- distribution is another bell-shaped (unimodal and symmetric) distribution, like the normal distribution; and the center of the t- distribution is standardized at zero, like the center of the standard normal distribution.

Like all distributions that are used as probability models, the normal and the t- distribution are both scaled, so the total area under each of them is 1.

So how is the t-distribution fundamentally different from the normal distribution?

• The spread .

The following picture illustrates the fundamental difference between the normal distribution and the t-distribution:

Here we have an image which illustrates the fundamental difference between the normal distribution and the t- distribution:

You can see in the picture that the t- distribution has slightly less area near the expected central value than the normal distribution does, and you can see that the t distribution has correspondingly more area in the “tails” than the normal distribution does. (It’s often said that the t- distribution has “fatter tails” or “heavier tails” than the normal distribution.)

This reflects the fact that the t- distribution has a larger spread than the normal distribution. The same total area of 1 is spread out over a slightly wider range on the t- distribution, making it a bit lower near the center compared to the normal distribution, and giving the t- distribution slightly more probability in the ‘tails’ compared to the normal distribution.

Therefore, the t- distribution ends up being the appropriate model in certain cases where there is more variability than would be predicted by the normal distribution. One of these cases is stock values, which have more variability (or “volatility,” to use the economic term) than would be predicted by the normal distribution.

There’s actually an entire family of t- distributions. They all have similar formulas (but the math is beyond the scope of this introductory course in statistics), and they all have slightly “fatter tails” than the normal distribution. But some are closer to normal than others.

The t- distributions that have higher “degrees of freedom” are closer to normal (degrees of freedom is a mathematical concept that we won’t study in this course, beyond merely mentioning it here). So, there’s a t- distribution “with one degree of freedom,” another t- distribution “with 2 degrees of freedom” which is slightly closer to normal, another t- distribution “with 3 degrees of freedom” which is a bit closer to normal than the previous ones, and so on.

The following picture illustrates this idea with just a couple of t- distributions (note that “degrees of freedom” is abbreviated “d.f.” on the picture):

The test statistic for our t-test for one population mean is a t -score which follows a t- distribution with (n – 1) degrees of freedom. Recall that each t- distribution is indexed according to “degrees of freedom.” Notice that, in the context of a test for a mean, the degrees of freedom depend on the sample size in the study.

Remember that we said that higher degrees of freedom indicate that the t- distribution is closer to normal. So in the context of a test for the mean, the larger the sample size , the higher the degrees of freedom, and the closer the t- distribution is to a normal z distribution .

As a result, in the context of a test for a mean, the effect of the t- distribution is most important for a study with a relatively small sample size .

We are now done introducing the t-distribution. What are implications of all of this?

• The null distribution of our t-test statistic is the t-distribution with (n-1) d.f. In other words, when Ho is true (i.e., when μ = μ 0 (mu = mu_zero)), our test statistic has a t-distribution with (n-1) d.f., and this is the distribution under which we find p-values.
• For a large sample size (n), the null distribution of the test statistic is approximately Z, so whether we use t(n – 1) or Z to calculate the p-values does not make a big difference.

Statistics Tutorial

Descriptive statistics, inferential statistics, stat reference, statistics - hypothesis testing.

Hypothesis testing is a formal way of checking if a hypothesis about a population is true or not.

A hypothesis is a claim about a population parameter .

A hypothesis test is a formal procedure to check if a hypothesis is true or not.

Examples of claims that can be checked:

The average height of people in Denmark is more than 170 cm.

The share of left handed people in Australia is not 10%.

The average income of dentists is less the average income of lawyers.

The Null and Alternative Hypothesis

Hypothesis testing is based on making two different claims about a population parameter.

The null hypothesis (\(H_{0} \)) and the alternative hypothesis (\(H_{1}\)) are the claims.

The two claims needs to be mutually exclusive , meaning only one of them can be true.

The alternative hypothesis is typically what we are trying to prove.

For example, we want to check the following claim:

"The average height of people in Denmark is more than 170 cm."

In this case, the parameter is the average height of people in Denmark (\(\mu\)).

The null and alternative hypothesis would be:

Null hypothesis : The average height of people in Denmark is 170 cm.

Alternative hypothesis : The average height of people in Denmark is more than 170 cm.

The claims are often expressed with symbols like this:

\(H_{0}\): \(\mu = 170 \: cm \)

\(H_{1}\): \(\mu > 170 \: cm \)

If the data supports the alternative hypothesis, we reject the null hypothesis and accept the alternative hypothesis.

If the data does not support the alternative hypothesis, we keep the null hypothesis.

Note: The alternative hypothesis is also referred to as (\(H_{A} \)).

The Significance Level

The significance level (\(\alpha\)) is the uncertainty we accept when rejecting the null hypothesis in the hypothesis test.

The significance level is a percentage probability of accidentally making the wrong conclusion.

Typical significance levels are:

• \(\alpha = 0.1\) (10%)
• \(\alpha = 0.05\) (5%)
• \(\alpha = 0.01\) (1%)

A lower significance level means that the evidence in the data needs to be stronger to reject the null hypothesis.

There is no "correct" significance level - it only states the uncertainty of the conclusion.

Note: A 5% significance level means that when we reject a null hypothesis:

We expect to reject a true null hypothesis 5 out of 100 times.

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The Test Statistic

The test statistic is used to decide the outcome of the hypothesis test.

The test statistic is a standardized value calculated from the sample.

Standardization means converting a statistic to a well known probability distribution .

The type of probability distribution depends on the type of test.

Common examples are:

• Standard Normal Distribution (Z): used for Testing Population Proportions
• Student's T-Distribution (T): used for Testing Population Means

Note: You will learn how to calculate the test statistic for each type of test in the following chapters.

The Critical Value and P-Value Approach

There are two main approaches used for hypothesis tests:

• The critical value approach compares the test statistic with the critical value of the significance level.
• The p-value approach compares the p-value of the test statistic and with the significance level.

The Critical Value Approach

The critical value approach checks if the test statistic is in the rejection region .

The rejection region is an area of probability in the tails of the distribution.

The size of the rejection region is decided by the significance level (\(\alpha\)).

The value that separates the rejection region from the rest is called the critical value .

Here is a graphical illustration:

If the test statistic is inside this rejection region, the null hypothesis is rejected .

For example, if the test statistic is 2.3 and the critical value is 2 for a significance level (\(\alpha = 0.05\)):

We reject the null hypothesis (\(H_{0} \)) at 0.05 significance level (\(\alpha\))

The P-Value Approach

The p-value approach checks if the p-value of the test statistic is smaller than the significance level (\(\alpha\)).

The p-value of the test statistic is the area of probability in the tails of the distribution from the value of the test statistic.

If the p-value is smaller than the significance level, the null hypothesis is rejected .

The p-value directly tells us the lowest significance level where we can reject the null hypothesis.

For example, if the p-value is 0.03:

We reject the null hypothesis (\(H_{0} \)) at a 0.05 significance level (\(\alpha\))

We keep the null hypothesis (\(H_{0}\)) at a 0.01 significance level (\(\alpha\))

Note: The two approaches are only different in how they present the conclusion.

Steps for a Hypothesis Test

The following steps are used for a hypothesis test:

• Check the conditions
• Define the claims
• Decide the significance level
• Calculate the test statistic

One condition is that the sample is randomly selected from the population.

The other conditions depends on what type of parameter you are testing the hypothesis for.

Common parameters to test hypotheses are:

• Proportions (for qualitative data)
• Mean values (for numerical data)

You will learn the steps for both types in the following pages.

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• Fundamental Analysis

Hypothesis to Be Tested: Definition and 4 Steps for Testing with Example

What Is Hypothesis Testing?

Hypothesis testing, sometimes called significance testing, is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis.

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Such data may come from a larger population, or from a data-generating process. The word "population" will be used for both of these cases in the following descriptions.

Key Takeaways

• Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data.
• The test provides evidence concerning the plausibility of the hypothesis, given the data.
• Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed.
• The four steps of hypothesis testing include stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.

How Hypothesis Testing Works

In hypothesis testing, an  analyst  tests a statistical sample, with the goal of providing evidence on the plausibility of the null hypothesis.

Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed. All analysts use a random population sample to test two different hypotheses: the null hypothesis and the alternative hypothesis.

The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a null hypothesis may state that the population mean return is equal to zero. The alternative hypothesis is effectively the opposite of a null hypothesis (e.g., the population mean return is not equal to zero). Thus, they are mutually exclusive , and only one can be true. However, one of the two hypotheses will always be true.

The null hypothesis is a statement about a population parameter, such as the population mean, that is assumed to be true.

4 Steps of Hypothesis Testing

All hypotheses are tested using a four-step process:

• The first step is for the analyst to state the hypotheses.
• The second step is to formulate an analysis plan, which outlines how the data will be evaluated.
• The third step is to carry out the plan and analyze the sample data.
• The final step is to analyze the results and either reject the null hypothesis, or state that the null hypothesis is plausible, given the data.

Real-World Example of Hypothesis Testing

If, for example, a person wants to test that a penny has exactly a 50% chance of landing on heads, the null hypothesis would be that 50% is correct, and the alternative hypothesis would be that 50% is not correct.

Mathematically, the null hypothesis would be represented as Ho: P = 0.5. The alternative hypothesis would be denoted as "Ha" and be identical to the null hypothesis, except with the equal sign struck-through, meaning that it does not equal 50%.

A random sample of 100 coin flips is taken, and the null hypothesis is then tested. If it is found that the 100 coin flips were distributed as 40 heads and 60 tails, the analyst would assume that a penny does not have a 50% chance of landing on heads and would reject the null hypothesis and accept the alternative hypothesis.

If, on the other hand, there were 48 heads and 52 tails, then it is plausible that the coin could be fair and still produce such a result. In cases such as this where the null hypothesis is "accepted," the analyst states that the difference between the expected results (50 heads and 50 tails) and the observed results (48 heads and 52 tails) is "explainable by chance alone."

Some staticians attribute the first hypothesis tests to satirical writer John Arbuthnot in 1710, who studied male and female births in England after observing that in nearly every year, male births exceeded female births by a slight proportion. Arbuthnot calculated that the probability of this happening by chance was small, and therefore it was due to “divine providence.”

What is Hypothesis Testing?

Hypothesis testing refers to a process used by analysts to assess the plausibility of a hypothesis by using sample data. In hypothesis testing, statisticians formulate two hypotheses: the null hypothesis and the alternative hypothesis. A null hypothesis determines there is no difference between two groups or conditions, while the alternative hypothesis determines that there is a difference. Researchers evaluate the statistical significance of the test based on the probability that the null hypothesis is true.

What are the Four Key Steps Involved in Hypothesis Testing?

Hypothesis testing begins with an analyst stating two hypotheses, with only one that can be right. The analyst then formulates an analysis plan, which outlines how the data will be evaluated. Next, they move to the testing phase and analyze the sample data. Finally, the analyst analyzes the results and either rejects the null hypothesis or states that the null hypothesis is plausible, given the data.

What are the Benefits of Hypothesis Testing?

Hypothesis testing helps assess the accuracy of new ideas or theories by testing them against data. This allows researchers to determine whether the evidence supports their hypothesis, helping to avoid false claims and conclusions. Hypothesis testing also provides a framework for decision-making based on data rather than personal opinions or biases. By relying on statistical analysis, hypothesis testing helps to reduce the effects of chance and confounding variables, providing a robust framework for making informed conclusions.

What are the Limitations of Hypothesis Testing?

Hypothesis testing relies exclusively on data and doesn’t provide a comprehensive understanding of the subject being studied. Additionally, the accuracy of the results depends on the quality of the available data and the statistical methods used. Inaccurate data or inappropriate hypothesis formulation may lead to incorrect conclusions or failed tests. Hypothesis testing can also lead to errors, such as analysts either accepting or rejecting a null hypothesis when they shouldn’t have. These errors may result in false conclusions or missed opportunities to identify significant patterns or relationships in the data.

The Bottom Line

Hypothesis testing refers to a statistical process that helps researchers and/or analysts determine the reliability of a study. By using a well-formulated hypothesis and set of statistical tests, individuals or businesses can make inferences about the population that they are studying and draw conclusions based on the data presented. There are different types of hypothesis testing, each with their own set of rules and procedures. However, all hypothesis testing methods have the same four step process, which includes stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result. Hypothesis testing plays a vital part of the scientific process, helping to test assumptions and make better data-based decisions.

Sage. " Introduction to Hypothesis Testing. " Page 4.

Elder Research. " Who Invented the Null Hypothesis? "

Formplus. " Hypothesis Testing: Definition, Uses, Limitations and Examples. "

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Teach yourself statistics

Hypothesis Test for a Mean

This lesson explains how to conduct a hypothesis test of a mean, when the following conditions are met:

• The sampling method is simple random sampling .
• The sampling distribution is normal or nearly normal.

Generally, the sampling distribution will be approximately normally distributed if any of the following conditions apply.

• The population distribution is normal.
• The population distribution is symmetric , unimodal , without outliers , and the sample size is 15 or less.
• The population distribution is moderately skewed , unimodal, without outliers, and the sample size is between 16 and 40.
• The sample size is greater than 40, without outliers.

This approach consists of four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results.

State the Hypotheses

Every hypothesis test requires the analyst to state a null hypothesis and an alternative hypothesis . The hypotheses are stated in such a way that they are mutually exclusive. That is, if one is true, the other must be false; and vice versa.

The table below shows three sets of hypotheses. Each makes a statement about how the population mean μ is related to a specified value M . (In the table, the symbol ≠ means " not equal to ".)

The first set of hypotheses (Set 1) is an example of a two-tailed test , since an extreme value on either side of the sampling distribution would cause a researcher to reject the null hypothesis. The other two sets of hypotheses (Sets 2 and 3) are one-tailed tests , since an extreme value on only one side of the sampling distribution would cause a researcher to reject the null hypothesis.

Formulate an Analysis Plan

The analysis plan describes how to use sample data to accept or reject the null hypothesis. It should specify the following elements.

• Significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.
• Test method. Use the one-sample t-test to determine whether the hypothesized mean differs significantly from the observed sample mean.

Analyze Sample Data

Using sample data, conduct a one-sample t-test. This involves finding the standard error, degrees of freedom, test statistic, and the P-value associated with the test statistic.

SE = s * sqrt{ ( 1/n ) * [ ( N - n ) / ( N - 1 ) ] }

SE = s / sqrt( n )

• Degrees of freedom. The degrees of freedom (DF) is equal to the sample size (n) minus one. Thus, DF = n - 1.

t = ( x - μ) / SE

• P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a t statistic, use the t Distribution Calculator to assess the probability associated with the t statistic, given the degrees of freedom computed above. (See sample problems at the end of this lesson for examples of how this is done.)

Sample Size Calculator

As you probably noticed, the process of hypothesis testing can be complex. When you need to test a hypothesis about a mean score, consider using the Sample Size Calculator. The calculator is fairly easy to use, and it is free. You can find the Sample Size Calculator in Stat Trek's main menu under the Stat Tools tab. Or you can tap the button below.

Interpret Results

If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level , and rejecting the null hypothesis when the P-value is less than the significance level.

Test Your Understanding

In this section, two sample problems illustrate how to conduct a hypothesis test of a mean score. The first problem involves a two-tailed test; the second problem, a one-tailed test.

Problem 1: Two-Tailed Test

An inventor has developed a new, energy-efficient lawn mower engine. He claims that the engine will run continuously for 5 hours (300 minutes) on a single gallon of regular gasoline. From his stock of 2000 engines, the inventor selects a simple random sample of 50 engines for testing. The engines run for an average of 295 minutes, with a standard deviation of 20 minutes. Test the null hypothesis that the mean run time is 300 minutes against the alternative hypothesis that the mean run time is not 300 minutes. Use a 0.05 level of significance. (Assume that run times for the population of engines are normally distributed.)

Solution: The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:

Null hypothesis: μ = 300

Alternative hypothesis: μ ≠ 300

• Formulate an analysis plan . For this analysis, the significance level is 0.05. The test method is a one-sample t-test .

SE = s / sqrt(n) = 20 / sqrt(50) = 20/7.07 = 2.83

DF = n - 1 = 50 - 1 = 49

t = ( x - μ) / SE = (295 - 300)/2.83 = -1.77

where s is the standard deviation of the sample, x is the sample mean, μ is the hypothesized population mean, and n is the sample size.

Since we have a two-tailed test , the P-value is the probability that the t statistic having 49 degrees of freedom is less than -1.77 or greater than 1.77. We use the t Distribution Calculator to find P(t < -1.77) is about 0.04.

• If you enter 1.77 as the sample mean in the t Distribution Calculator, you will find the that the P(t < 1.77) is about 0.04. Therefore, P(t >  1.77) is 1 minus 0.96 or 0.04. Thus, the P-value = 0.04 + 0.04 = 0.08.
• Interpret results . Since the P-value (0.08) is greater than the significance level (0.05), we cannot reject the null hypothesis.

Note: If you use this approach on an exam, you may also want to mention why this approach is appropriate. Specifically, the approach is appropriate because the sampling method was simple random sampling, the population was normally distributed, and the sample size was small relative to the population size (less than 5%).

Problem 2: One-Tailed Test

Bon Air Elementary School has 1000 students. The principal of the school thinks that the average IQ of students at Bon Air is at least 110. To prove her point, she administers an IQ test to 20 randomly selected students. Among the sampled students, the average IQ is 108 with a standard deviation of 10. Based on these results, should the principal accept or reject her original hypothesis? Assume a significance level of 0.01. (Assume that test scores in the population of engines are normally distributed.)

Null hypothesis: μ >= 110

Alternative hypothesis: μ < 110

• Formulate an analysis plan . For this analysis, the significance level is 0.01. The test method is a one-sample t-test .

SE = s / sqrt(n) = 10 / sqrt(20) = 10/4.472 = 2.236

DF = n - 1 = 20 - 1 = 19

t = ( x - μ) / SE = (108 - 110)/2.236 = -0.894

Here is the logic of the analysis: Given the alternative hypothesis (μ < 110), we want to know whether the observed sample mean is small enough to cause us to reject the null hypothesis.

The observed sample mean produced a t statistic test statistic of -0.894. We use the t Distribution Calculator to find P(t < -0.894) is about 0.19.

• This means we would expect to find a sample mean of 108 or smaller in 19 percent of our samples, if the true population IQ were 110. Thus the P-value in this analysis is 0.19.
• Interpret results . Since the P-value (0.19) is greater than the significance level (0.01), we cannot reject the null hypothesis.

Step 2: Collect data. For a statistical test to be valid, it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in. Hypothesis testing example.

Step 2: State the Alternate Hypothesis. The claim is that the students have above average IQ scores, so: H 1: μ > 100. The fact that we are looking for scores "greater than" a certain point means that this is a one-tailed test. Step 3: Draw a picture to help you visualize the problem. Step 4: State the alpha level.

Test Statistic: z = x¯¯¯ −μo σ/ n−−√ z = x ¯ − μ o σ / n since it is calculated as part of the testing of the hypothesis. Definition 7.1.4 7.1. 4. p - value: probability that the test statistic will take on more extreme values than the observed test statistic, given that the null hypothesis is true.

Hypothesis testing is a crucial procedure to perform when you want to make inferences about a population using a random sample. These inferences include estimating population properties such as the mean, differences between means, proportions, and the relationships between variables. This post provides an overview of statistical hypothesis testing.

Testing Hypotheses using Confidence Intervals. We can start the evaluation of the hypothesis setup by comparing 2006 and 2012 run times using a point estimate from the 2012 sample: x¯12 = 95.61 x ¯ 12 = 95.61 minutes. This estimate suggests the average time is actually longer than the 2006 time, 93.29 minutes.

A hypothesis test consists of five steps: 1. State the hypotheses. State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false. 2. Determine a significance level to use for the hypothesis. Decide on a significance level.

In hypothesis testing, the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis.The null hypothesis is usually denoted \(H_0\) while the alternative hypothesis is usually denoted \(H_1\). An hypothesis test is a statistical decision; the conclusion will either be to reject the null hypothesis in favor ...

The general idea of hypothesis testing involves: Making an initial assumption. Collecting evidence (data). Based on the available evidence (data), deciding whether to reject or not reject the initial assumption. Every hypothesis test — regardless of the population parameter involved — requires the above three steps.

Photo from StepUp Analytics. Hypothesis testing is a method of statistical inference that considers the null hypothesis H₀ vs. the alternative hypothesis Ha, where we are typically looking to assess evidence against H₀. Such a test is used to compare data sets against one another, or compare a data set against some external standard. The former being a two sample test (independent or ...

Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values to make conclusions about hypotheses. Simple ...

Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence.

Hypothesis Testing Calculator. The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is ...

Statistical tests are used in hypothesis testing. They can be used to: determine whether a predictor variable has a statistically significant relationship with an outcome variable. estimate the difference between two or more groups. Statistical tests assume a null hypothesis of no relationship or no difference between groups. Then they ...

Then, if the null hypothesis is wrong, then the data will tend to group at a point that is not the value in the null hypothesis (1.2), and then our p-value will wind up being very small. If the null hypothesis is correct, or close to being correct, then the p-value will be larger, because the data values will group around the value we hypothesized.

The Four Steps in Hypothesis Testing. STEP 1: State the appropriate null and alternative hypotheses, Ho and Ha. STEP 2: Obtain a random sample, collect relevant data, and check whether the data meet the conditions under which the test can be used. If the conditions are met, summarize the data using a test statistic.

Hypothesis testing is based on making two different claims about a population parameter. The null hypothesis ( H 0) and the alternative hypothesis ( H 1) are the claims. The two claims needs to be mutually exclusive, meaning only one of them can be true. The alternative hypothesis is typically what we are trying to prove.

To use a test statistic to evaluate statistical significance, you either compare it to a critical value or use it to calculate the p-value. Statisticians named the hypothesis tests after the test statistics because they're the quantity that the tests actually evaluate. For example, t-tests assess t-values, F-tests evaluate F-values, and chi ...

To find the p value for your sample, do the following: Identify the correct test statistic. Calculate the test statistic using the relevant properties of your sample. Specify the characteristics of the test statistic's sampling distribution. Place your test statistic in the sampling distribution to find the p value.

Hypothesis testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used ...

Hypothesis testing is a vital process in inferential statistics where the goal is to use sample data to draw conclusions about an entire population. In the testing process, you use significance levels and p-values to determine whether the test results are statistically significant. You hear about results being statistically significant all of ...

The test statistic is a number calculated from a statistical test of a hypothesis. It shows how closely your observed data match the distribution expected under the null hypothesis of that statistical test. The test statistic is used to calculate the p value of your results, helping to decide whether to reject your null hypothesis.

Solution: The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below: State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.