the goal of a hypothesis test is to

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

Featured Posts

Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Join the Statology Community

Sign up to receive Statology's exclusive study resource: 100 practice problems with step-by-step solutions. Plus, get our latest insights, tutorials, and data analysis tips straight to your inbox!

By subscribing you accept Statology's Privacy Policy.

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Unit 12: Significance tests (hypothesis testing)

About this unit.

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.

The idea of significance tests

• Simple hypothesis testing (Opens a modal)
• Idea behind hypothesis testing (Opens a modal)
• Examples of null and alternative hypotheses (Opens a modal)
• P-values and significance tests (Opens a modal)
• Comparing P-values to different significance levels (Opens a modal)
• Estimating a P-value from a simulation (Opens a modal)
• Using P-values to make conclusions (Opens a modal)
• Simple hypothesis testing Get 3 of 4 questions to level up!
• Writing null and alternative hypotheses Get 3 of 4 questions to level up!
• Estimating P-values from simulations Get 3 of 4 questions to level up!

Error probabilities and power

• Introduction to Type I and Type II errors (Opens a modal)
• Type 1 errors (Opens a modal)
• Examples identifying Type I and Type II errors (Opens a modal)
• Introduction to power in significance tests (Opens a modal)
• Examples thinking about power in significance tests (Opens a modal)
• Consequences of errors and significance (Opens a modal)
• Type I vs Type II error Get 3 of 4 questions to level up!
• Error probabilities and power Get 3 of 4 questions to level up!

Tests about a population proportion

• Constructing hypotheses for a significance test about a proportion (Opens a modal)
• Conditions for a z test about a proportion (Opens a modal)
• Reference: Conditions for inference on a proportion (Opens a modal)
• Calculating a z statistic in a test about a proportion (Opens a modal)
• Calculating a P-value given a z statistic (Opens a modal)
• Making conclusions in a test about a proportion (Opens a modal)
• Writing hypotheses for a test about a proportion Get 3 of 4 questions to level up!
• Conditions for a z test about a proportion Get 3 of 4 questions to level up!
• Calculating the test statistic in a z test for a proportion Get 3 of 4 questions to level up!
• Calculating the P-value in a z test for a proportion Get 3 of 4 questions to level up!
• Making conclusions in a z test for a proportion Get 3 of 4 questions to level up!

Tests about a population mean

• Writing hypotheses for a significance test about a mean (Opens a modal)
• Conditions for a t test about a mean (Opens a modal)
• Reference: Conditions for inference on a mean (Opens a modal)
• When to use z or t statistics in significance tests (Opens a modal)
• Example calculating t statistic for a test about a mean (Opens a modal)
• Using TI calculator for P-value from t statistic (Opens a modal)
• Using a table to estimate P-value from t statistic (Opens a modal)
• Comparing P-value from t statistic to significance level (Opens a modal)
• Free response example: Significance test for a mean (Opens a modal)
• Writing hypotheses for a test about a mean Get 3 of 4 questions to level up!
• Conditions for a t test about a mean Get 3 of 4 questions to level up!
• Calculating the test statistic in a t test for a mean Get 3 of 4 questions to level up!
• Calculating the P-value in a t test for a mean Get 3 of 4 questions to level up!
• Making conclusions in a t test for a mean Get 3 of 4 questions to level up!

More significance testing videos

• Hypothesis testing and p-values (Opens a modal)
• One-tailed and two-tailed tests (Opens a modal)
• Z-statistics vs. T-statistics (Opens a modal)
• Small sample hypothesis test (Opens a modal)
• Large sample proportion hypothesis testing (Opens a modal)

• Search Search Please fill out this field.

What Is Hypothesis Testing?

• How It Works

4 Step Process

The bottom line.

• Fundamental Analysis

Hypothesis Testing: 4 Steps and Example

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 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, intending to provide evidence on the plausibility of the null hypothesis. Statistical analysts measure and examine 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. 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.

• State the hypotheses.
• Formulate an analysis plan, which outlines how the data will be evaluated.
• Carry out the plan and analyze the sample data.
• Analyze the results and either reject the null hypothesis, or state that the null hypothesis is plausible, given the data.

Example of Hypothesis Testing

If an individual 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 is represented as Ho: P = 0.5. The alternative hypothesis is shown as "Ha" and is 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 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 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."

When Did Hypothesis Testing Begin?

Some statisticians 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 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.

Hypothesis testing refers to a statistical process that helps researchers 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. 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.

Sage. " Introduction to Hypothesis Testing ," Page 4.

Elder Research. " Who Invented the Null Hypothesis? "

Formplus. " Hypothesis Testing: Definition, Uses, Limitations and Examples ."

• Terms of Service
• Editorial Policy
• Privacy Policy
• Your Privacy Choices

• Hypothesis Testing: Definition, Uses, Limitations + Examples

Hypothesis testing is as old as the scientific method and is at the heart of the research process. 

Research exists to validate or disprove assumptions about various phenomena. The process of validation involves testing and it is in this context that we will explore hypothesis testing. 

What is a Hypothesis? 

A hypothesis is a calculated prediction or assumption about a population parameter based on limited evidence. The whole idea behind hypothesis formulation is testing—this means the researcher subjects his or her calculated assumption to a series of evaluations to know whether they are true or false. 

Typically, every research starts with a hypothesis—the investigator makes a claim and experiments to prove that this claim is true or false . For instance, if you predict that students who drink milk before class perform better than those who don’t, then this becomes a hypothesis that can be confirmed or refuted using an experiment.  

Read: What is Empirical Research Study? [Examples & Method]

What are the Types of Hypotheses? 

1. simple hypothesis.

Also known as a basic hypothesis, a simple hypothesis suggests that an independent variable is responsible for a corresponding dependent variable. In other words, an occurrence of the independent variable inevitably leads to an occurrence of the dependent variable. 

Typically, simple hypotheses are considered as generally true, and they establish a causal relationship between two variables. 

Examples of Simple Hypothesis  

• Drinking soda and other sugary drinks can cause obesity. 
• Smoking cigarettes daily leads to lung cancer.

2. Complex Hypothesis

A complex hypothesis is also known as a modal. It accounts for the causal relationship between two independent variables and the resulting dependent variables. This means that the combination of the independent variables leads to the occurrence of the dependent variables . 

Examples of Complex Hypotheses  

• Adults who do not smoke and drink are less likely to develop liver-related conditions.
• Global warming causes icebergs to melt which in turn causes major changes in weather patterns.

3. Null Hypothesis

As the name suggests, a null hypothesis is formed when a researcher suspects that there’s no relationship between the variables in an observation. In this case, the purpose of the research is to approve or disapprove this assumption. 

Examples of Null Hypothesis

• This is no significant change in a student’s performance if they drink coffee or tea before classes. 
• There’s no significant change in the growth of a plant if one uses distilled water only or vitamin-rich water. 

Read: Research Report: Definition, Types + [Writing Guide]

4. Alternative Hypothesis 

To disapprove a null hypothesis, the researcher has to come up with an opposite assumption—this assumption is known as the alternative hypothesis. This means if the null hypothesis says that A is false, the alternative hypothesis assumes that A is true. 

An alternative hypothesis can be directional or non-directional depending on the direction of the difference. A directional alternative hypothesis specifies the direction of the tested relationship, stating that one variable is predicted to be larger or smaller than the null value while a non-directional hypothesis only validates the existence of a difference without stating its direction. 

Examples of Alternative Hypotheses  

• Starting your day with a cup of tea instead of a cup of coffee can make you more alert in the morning. 
• The growth of a plant improves significantly when it receives distilled water instead of vitamin-rich water. 

5. Logical Hypothesis

Logical hypotheses are some of the most common types of calculated assumptions in systematic investigations. It is an attempt to use your reasoning to connect different pieces in research and build a theory using little evidence. In this case, the researcher uses any data available to him, to form a plausible assumption that can be tested. 

Examples of Logical Hypothesis

• Waking up early helps you to have a more productive day. 
• Beings from Mars would not be able to breathe the air in the atmosphere of the Earth. 

6. Empirical Hypothesis  

After forming a logical hypothesis, the next step is to create an empirical or working hypothesis. At this stage, your logical hypothesis undergoes systematic testing to prove or disprove the assumption. An empirical hypothesis is subject to several variables that can trigger changes and lead to specific outcomes. 

Examples of Empirical Testing 

• People who eat more fish run faster than people who eat meat.
• Women taking vitamin E grow hair faster than those taking vitamin K.

7. Statistical Hypothesis

When forming a statistical hypothesis, the researcher examines the portion of a population of interest and makes a calculated assumption based on the data from this sample. A statistical hypothesis is most common with systematic investigations involving a large target audience. Here, it’s impossible to collect responses from every member of the population so you have to depend on data from your sample and extrapolate the results to the wider population. 

Examples of Statistical Hypothesis  

• 45% of students in Louisiana have middle-income parents. 
• 80% of the UK’s population gets a divorce because of irreconcilable differences.

What is Hypothesis Testing? 

Hypothesis testing is an assessment method that allows researchers to determine the plausibility of a hypothesis. It involves testing an assumption about a specific population parameter to know whether it’s true or false. These population parameters include variance, standard deviation, and median. 

Typically, hypothesis testing starts with developing a null hypothesis and then performing several tests that support or reject the null hypothesis. The researcher uses test statistics to compare the association or relationship between two or more variables. 

Explore: Research Bias: Definition, Types + Examples

Researchers also use hypothesis testing to calculate the coefficient of variation and determine if the regression relationship and the correlation coefficient are statistically significant.

How Hypothesis Testing Works

The basis of hypothesis testing is to examine and analyze the null hypothesis and alternative hypothesis to know which one is the most plausible assumption. Since both assumptions are mutually exclusive, only one can be true. In other words, the occurrence of a null hypothesis destroys the chances of the alternative coming to life, and vice-versa. 

Interesting: 21 Chrome Extensions for Academic Researchers in 2021

What Are The Stages of Hypothesis Testing?  

To successfully confirm or refute an assumption, the researcher goes through five (5) stages of hypothesis testing; 

• Determine the null hypothesis
• Specify the alternative hypothesis
• Set the significance level
• Calculate the test statistics and corresponding P-value
• Draw your conclusion
• Determine the Null Hypothesis

Like we mentioned earlier, hypothesis testing starts with creating a null hypothesis which stands as an assumption that a certain statement is false or implausible. For example, the null hypothesis (H0) could suggest that different subgroups in the research population react to a variable in the same way. 

• Specify the Alternative Hypothesis

Once you know the variables for the null hypothesis, the next step is to determine the alternative hypothesis. The alternative hypothesis counters the null assumption by suggesting the statement or assertion is true. Depending on the purpose of your research, the alternative hypothesis can be one-sided or two-sided. 

Using the example we established earlier, the alternative hypothesis may argue that the different sub-groups react differently to the same variable based on several internal and external factors. 

• Set the Significance Level

Many researchers create a 5% allowance for accepting the value of an alternative hypothesis, even if the value is untrue. This means that there is a 0.05 chance that one would go with the value of the alternative hypothesis, despite the truth of the null hypothesis. 

Something to note here is that the smaller the significance level, the greater the burden of proof needed to reject the null hypothesis and support the alternative hypothesis.

Explore: What is Data Interpretation? + [Types, Method & Tools]

• Calculate the Test Statistics and Corresponding P-Value 

Test statistics in hypothesis testing allow you to compare different groups between variables while the p-value accounts for the probability of obtaining sample statistics if your null hypothesis is true. In this case, your test statistics can be the mean, median and similar parameters. 

If your p-value is 0.65, for example, then it means that the variable in your hypothesis will happen 65 in100 times by pure chance. Use this formula to determine the p-value for your data: 

• Draw Your Conclusions

After conducting a series of tests, you should be able to agree or refute the hypothesis based on feedback and insights from your sample data.  

Applications of Hypothesis Testing in Research

Hypothesis testing isn’t only confined to numbers and calculations; it also has several real-life applications in business, manufacturing, advertising, and medicine. 

In a factory or other manufacturing plants, hypothesis testing is an important part of quality and production control before the final products are approved and sent out to the consumer. 

During ideation and strategy development, C-level executives use hypothesis testing to evaluate their theories and assumptions before any form of implementation. For example, they could leverage hypothesis testing to determine whether or not some new advertising campaign, marketing technique, etc. causes increased sales. 

In addition, hypothesis testing is used during clinical trials to prove the efficacy of a drug or new medical method before its approval for widespread human usage. 

What is an Example of Hypothesis Testing?

An employer claims that her workers are of above-average intelligence. She takes a random sample of 20 of them and gets the following results: 

Mean IQ Scores: 110

Standard Deviation: 15 

Mean Population IQ: 100

Step 1: Using the value of the mean population IQ, we establish the null hypothesis as 100.

Step 2: State that the alternative hypothesis is greater than 100.

Step 3: State the alpha level as 0.05 or 5% 

Step 4: Find the rejection region area (given by your alpha level above) from the z-table. An area of .05 is equal to a z-score of 1.645.

Step 5: Calculate the test statistics using this formula

Z = (110–100) ÷ (15÷√20) 

10 ÷ 3.35 = 2.99 

If the value of the test statistics is higher than the value of the rejection region, then you should reject the null hypothesis. If it is less, then you cannot reject the null. 

In this case, 2.99 > 1.645 so we reject the null. 

Importance/Benefits of Hypothesis Testing 

The most significant benefit of hypothesis testing is it allows you to evaluate the strength of your claim or assumption before implementing it in your data set. Also, hypothesis testing is the only valid method to prove that something “is or is not”. Other benefits include: 

• Hypothesis testing provides a reliable framework for making any data decisions for your population of interest. 
• It helps the researcher to successfully extrapolate data from the sample to the larger population. 
• Hypothesis testing allows the researcher to determine whether the data from the sample is statistically significant. 
• Hypothesis testing is one of the most important processes for measuring the validity and reliability of outcomes in any systematic investigation. 
• It helps to provide links to the underlying theory and specific research questions.

Criticism and Limitations of Hypothesis Testing

Several limitations of hypothesis testing can affect the quality of data you get from this process. Some of these limitations include: 

• The interpretation of a p-value for observation depends on the stopping rule and definition of multiple comparisons. This makes it difficult to calculate since the stopping rule is subject to numerous interpretations, plus “multiple comparisons” are unavoidably ambiguous. 
• Conceptual issues often arise in hypothesis testing, especially if the researcher merges Fisher and Neyman-Pearson’s methods which are conceptually distinct. 
• In an attempt to focus on the statistical significance of the data, the researcher might ignore the estimation and confirmation by repeated experiments.
• Hypothesis testing can trigger publication bias, especially when it requires statistical significance as a criterion for publication.
• When used to detect whether a difference exists between groups, hypothesis testing can trigger absurd assumptions that affect the reliability of your observation.

Connect to Formplus, Get Started Now - It's Free!

• alternative hypothesis
• alternative vs null hypothesis
• complex hypothesis
• empirical hypothesis
• hypothesis testing
• logical hypothesis
• simple hypothesis
• statistical hypothesis
• busayo.longe

You may also like:

Internal Validity in Research: Definition, Threats, Examples

In this article, we will discuss the concept of internal validity, some clear examples, its importance, and how to test it.

Alternative vs Null Hypothesis: Pros, Cons, Uses & Examples

We are going to discuss alternative hypotheses and null hypotheses in this post and how they work in research.

What is Pure or Basic Research? + [Examples & Method]

Simple guide on pure or basic research, its methods, characteristics, advantages, and examples in science, medicine, education and psychology

Type I vs Type II Errors: Causes, Examples & Prevention

This article will discuss the two different types of errors in hypothesis testing and how you can prevent them from occurring in your research

Formplus - For Seamless Data Collection

Collect data the right way with a versatile data collection tool. try formplus and transform your work productivity today..

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons

Margin Size

• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

4.4: Hypothesis Testing

• Last updated
• Save as PDF
• Page ID 283

• David Diez, Christopher Barr, & Mine Çetinkaya-Rundel
• OpenIntro Statistics

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

Is the typical US runner getting faster or slower over time? We consider this question in the context of the Cherry Blossom Run, comparing runners in 2006 and 2012. Technological advances in shoes, training, and diet might suggest runners would be faster in 2012. An opposing viewpoint might say that with the average body mass index on the rise, people tend to run slower. In fact, all of these components might be influencing run time.

In addition to considering run times in this section, we consider a topic near and dear to most students: sleep. A recent study found that college students average about 7 hours of sleep per night.15 However, researchers at a rural college are interested in showing that their students sleep longer than seven hours on average. We investigate this topic in Section 4.3.4.

Hypothesis Testing Framework

The average time for all runners who finished the Cherry Blossom Run in 2006 was 93.29 minutes (93 minutes and about 17 seconds). We want to determine if the run10Samp data set provides strong evidence that the participants in 2012 were faster or slower than those runners in 2006, versus the other possibility that there has been no change. 16 We simplify these three options into two competing hypotheses :

• H 0 : The average 10 mile run time was the same for 2006 and 2012.
• H A : The average 10 mile run time for 2012 was different than that of 2006.

We call H 0 the null hypothesis and H A the alternative hypothesis.

Null and alternative hypotheses

• The null hypothesis (H 0 ) often represents either a skeptical perspective or a claim to be tested.
• The alternative hypothesis (H A ) represents an alternative claim under consideration and is often represented by a range of possible parameter values.

15 theloquitur.com/?p=1161

16 While we could answer this question by examining the entire population data (run10), we only consider the sample data (run10Samp), which is more realistic since we rarely have access to population data.

The null hypothesis often represents a skeptical position or a perspective of no difference. The alternative hypothesis often represents a new perspective, such as the possibility that there has been a change.

Hypothesis testing framework

The skeptic will not reject the null hypothesis (H 0 ), unless the evidence in favor of the alternative hypothesis (H A ) is so strong that she rejects H 0 in favor of H A .

The hypothesis testing framework is a very general tool, and we often use it without a second thought. If a person makes a somewhat unbelievable claim, we are initially skeptical. However, if there is sufficient evidence that supports the claim, we set aside our skepticism and reject the null hypothesis in favor of the alternative. The hallmarks of hypothesis testing are also found in the US court system.

Exercise \(\PageIndex{1}\)

A US court considers two possible claims about a defendant: she is either innocent or guilty. If we set these claims up in a hypothesis framework, which would be the null hypothesis and which the alternative? 17

Jurors examine the evidence to see whether it convincingly shows a defendant is guilty. Even if the jurors leave unconvinced of guilt beyond a reasonable doubt, this does not mean they believe the defendant is innocent. This is also the case with hypothesis testing: even if we fail to reject the null hypothesis, we typically do not accept the null hypothesis as true. Failing to find strong evidence for the alternative hypothesis is not equivalent to accepting the null hypothesis.

17 H 0 : The average cost is $650 per month, \(\mu\) = $650.

In the example with the Cherry Blossom Run, the null hypothesis represents no difference in the average time from 2006 to 2012. The alternative hypothesis represents something new or more interesting: there was a difference, either an increase or a decrease. These hypotheses can be described in mathematical notation using \(\mu_{12}\) as the average run time for 2012:

• H 0 : \(\mu_{12} = 93.29\)
• H A : \(\mu_{12} \ne 93.29\)

where 93.29 minutes (93 minutes and about 17 seconds) is the average 10 mile time for all runners in the 2006 Cherry Blossom Run. Using this mathematical notation, the hypotheses can now be evaluated using statistical tools. We call 93.29 the null value since it represents the value of the parameter if the null hypothesis is true. We will use the run10Samp data set to evaluate the hypothesis test.

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: \(\bar {x}_{12} = 95.61\) minutes. This estimate suggests the average time is actually longer than the 2006 time, 93.29 minutes. However, to evaluate whether this provides strong evidence that there has been a change, we must consider the uncertainty associated with \(\bar {x}_{12}\).

1 6 The jury considers whether the evidence is so convincing (strong) that there is no reasonable doubt regarding the person's guilt; in such a case, the jury rejects innocence (the null hypothesis) and concludes the defendant is guilty (alternative hypothesis).

We learned in Section 4.1 that there is fluctuation from one sample to another, and it is very unlikely that the sample mean will be exactly equal to our parameter; we should not expect \(\bar {x}_{12}\) to exactly equal \(\mu_{12}\). Given that \(\bar {x}_{12} = 95.61\), it might still be possible that the population average in 2012 has remained unchanged from 2006. The difference between \(\bar {x}_{12}\) and 93.29 could be due to sampling variation, i.e. the variability associated with the point estimate when we take a random sample.

In Section 4.2, confidence intervals were introduced as a way to find a range of plausible values for the population mean. Based on run10Samp, a 95% confidence interval for the 2012 population mean, \(\mu_{12}\), was calculated as

\[(92.45, 98.77)\]

Because the 2006 mean, 93.29, falls in the range of plausible values, we cannot say the null hypothesis is implausible. That is, we failed to reject the null hypothesis, H 0 .

Double negatives can sometimes be used in statistics

In many statistical explanations, we use double negatives. For instance, we might say that the null hypothesis is not implausible or we failed to reject the null hypothesis. Double negatives are used to communicate that while we are not rejecting a position, we are also not saying it is correct.

Example \(\PageIndex{1}\)

Next consider whether there is strong evidence that the average age of runners has changed from 2006 to 2012 in the Cherry Blossom Run. In 2006, the average age was 36.13 years, and in the 2012 run10Samp data set, the average was 35.05 years with a standard deviation of 8.97 years for 100 runners.

First, set up the hypotheses:

• H 0 : The average age of runners has not changed from 2006 to 2012, \(\mu_{age} = 36.13.\)
• H A : The average age of runners has changed from 2006 to 2012, \(\mu _{age} 6 \ne 36.13.\)

We have previously veri ed conditions for this data set. The normal model may be applied to \(\bar {y}\) and the estimate of SE should be very accurate. Using the sample mean and standard error, we can construct a 95% con dence interval for \(\mu _{age}\) to determine if there is sufficient evidence to reject H 0 :

\[\bar{y} \pm 1.96 \times \dfrac {s}{\sqrt {100}} \rightarrow 35.05 \pm 1.96 \times 0.90 \rightarrow (33.29, 36.81)\]

This confidence interval contains the null value, 36.13. Because 36.13 is not implausible, we cannot reject the null hypothesis. We have not found strong evidence that the average age is different than 36.13 years.

Exercise \(\PageIndex{2}\)

Colleges frequently provide estimates of student expenses such as housing. A consultant hired by a community college claimed that the average student housing expense was $650 per month. What are the null and alternative hypotheses to test whether this claim is accurate? 18

H A : The average cost is different than $650 per month, \(\mu \ne\) $650.

18 Applying the normal model requires that certain conditions are met. Because the data are a simple random sample and the sample (presumably) represents no more than 10% of all students at the college, the observations are independent. The sample size is also sufficiently large (n = 75) and the data exhibit only moderate skew. Thus, the normal model may be applied to the sample mean.

Exercise \(\PageIndex{3}\)

The community college decides to collect data to evaluate the $650 per month claim. They take a random sample of 75 students at their school and obtain the data represented in Figure 4.11. Can we apply the normal model to the sample mean?

If the court makes a Type 1 Error, this means the defendant is innocent (H 0 true) but wrongly convicted. A Type 2 Error means the court failed to reject H 0 (i.e. failed to convict the person) when she was in fact guilty (H A true).

Example \(\PageIndex{2}\)

The sample mean for student housing is $611.63 and the sample standard deviation is $132.85. Construct a 95% confidence interval for the population mean and evaluate the hypotheses of Exercise 4.22.

The standard error associated with the mean may be estimated using the sample standard deviation divided by the square root of the sample size. Recall that n = 75 students were sampled.

\[ SE = \dfrac {s}{\sqrt {n}} = \dfrac {132.85}{\sqrt {75}} = 15.34\]

You showed in Exercise 4.23 that the normal model may be applied to the sample mean. This ensures a 95% confidence interval may be accurately constructed:

\[\bar {x} \pm z*SE \rightarrow 611.63 \pm 1.96 \times 15.34 \times (581.56, 641.70)\]

Because the null value $650 is not in the confidence interval, a true mean of $650 is implausible and we reject the null hypothesis. The data provide statistically significant evidence that the actual average housing expense is less than $650 per month.

Decision Errors

Hypothesis tests are not flawless. Just think of the court system: innocent people are sometimes wrongly convicted and the guilty sometimes walk free. Similarly, we can make a wrong decision in statistical hypothesis tests. However, the difference is that we have the tools necessary to quantify how often we make such errors.

There are two competing hypotheses: the null and the alternative. In a hypothesis test, we make a statement about which one might be true, but we might choose incorrectly. There are four possible scenarios in a hypothesis test, which are summarized in Table 4.12.

A Type 1 Error is rejecting the null hypothesis when H0 is actually true. A Type 2 Error is failing to reject the null hypothesis when the alternative is actually true.

Exercise 4.25

In a US court, the defendant is either innocent (H 0 ) or guilty (H A ). What does a Type 1 Error represent in this context? What does a Type 2 Error represent? Table 4.12 may be useful.

To lower the Type 1 Error rate, we might raise our standard for conviction from "beyond a reasonable doubt" to "beyond a conceivable doubt" so fewer people would be wrongly convicted. However, this would also make it more difficult to convict the people who are actually guilty, so we would make more Type 2 Errors.

Exercise 4.26

How could we reduce the Type 1 Error rate in US courts? What influence would this have on the Type 2 Error rate?

To lower the Type 2 Error rate, we want to convict more guilty people. We could lower the standards for conviction from "beyond a reasonable doubt" to "beyond a little doubt". Lowering the bar for guilt will also result in more wrongful convictions, raising the Type 1 Error rate.

Exercise 4.27

How could we reduce the Type 2 Error rate in US courts? What influence would this have on the Type 1 Error rate?

A skeptic would have no reason to believe that sleep patterns at this school are different than the sleep patterns at another school.

Exercises 4.25-4.27 provide an important lesson:

If we reduce how often we make one type of error, we generally make more of the other type.

Hypothesis testing is built around rejecting or failing to reject the null hypothesis. That is, we do not reject H 0 unless we have strong evidence. But what precisely does strong evidence mean? As a general rule of thumb, for those cases where the null hypothesis is actually true, we do not want to incorrectly reject H 0 more than 5% of the time. This corresponds to a significance level of 0.05. We often write the significance level using \(\alpha\) (the Greek letter alpha): \(\alpha = 0.05.\) We discuss the appropriateness of different significance levels in Section 4.3.6.

If we use a 95% confidence interval to test a hypothesis where the null hypothesis is true, we will make an error whenever the point estimate is at least 1.96 standard errors away from the population parameter. This happens about 5% of the time (2.5% in each tail). Similarly, using a 99% con dence interval to evaluate a hypothesis is equivalent to a significance level of \(\alpha = 0.01\).

A confidence interval is, in one sense, simplistic in the world of hypothesis tests. Consider the following two scenarios:

• The null value (the parameter value under the null hypothesis) is in the 95% confidence interval but just barely, so we would not reject H 0 . However, we might like to somehow say, quantitatively, that it was a close decision.
• The null value is very far outside of the interval, so we reject H 0 . However, we want to communicate that, not only did we reject the null hypothesis, but it wasn't even close. Such a case is depicted in Figure 4.13.

In Section 4.3.4, we introduce a tool called the p-value that will be helpful in these cases. The p-value method also extends to hypothesis tests where con dence intervals cannot be easily constructed or applied.

Formal Testing using p-Values

The p-value is a way of quantifying the strength of the evidence against the null hypothesis and in favor of the alternative. Formally the p-value is a conditional probability.

definition: p-value

The p-value is the probability of observing data at least as favorable to the alternative hypothesis as our current data set, if the null hypothesis is true. We typically use a summary statistic of the data, in this chapter the sample mean, to help compute the p-value and evaluate the hypotheses.

A poll by the National Sleep Foundation found that college students average about 7 hours of sleep per night. Researchers at a rural school are interested in showing that students at their school sleep longer than seven hours on average, and they would like to demonstrate this using a sample of students. What would be an appropriate skeptical position for this research?

This is entirely based on the interests of the researchers. Had they been only interested in the opposite case - showing that their students were actually averaging fewer than seven hours of sleep but not interested in showing more than 7 hours - then our setup would have set the alternative as \(\mu < 7\).

We can set up the null hypothesis for this test as a skeptical perspective: the students at this school average 7 hours of sleep per night. The alternative hypothesis takes a new form reflecting the interests of the research: the students average more than 7 hours of sleep. We can write these hypotheses as

• H 0 : \(\mu\) = 7.
• H A : \(\mu\) > 7.

Using \(\mu\) > 7 as the alternative is an example of a one-sided hypothesis test. In this investigation, there is no apparent interest in learning whether the mean is less than 7 hours. (The standard error can be estimated from the sample standard deviation and the sample size: \(SE_{\bar {x}} = \dfrac {s_x}{\sqrt {n}} = \dfrac {1.75}{\sqrt {110}} = 0.17\)). Earlier we encountered a two-sided hypothesis where we looked for any clear difference, greater than or less than the null value.

Always use a two-sided test unless it was made clear prior to data collection that the test should be one-sided. Switching a two-sided test to a one-sided test after observing the data is dangerous because it can inflate the Type 1 Error rate.

TIP: One-sided and two-sided tests

If the researchers are only interested in showing an increase or a decrease, but not both, use a one-sided test. If the researchers would be interested in any difference from the null value - an increase or decrease - then the test should be two-sided.

TIP: Always write the null hypothesis as an equality

We will find it most useful if we always list the null hypothesis as an equality (e.g. \(\mu\) = 7) while the alternative always uses an inequality (e.g. \(\mu \ne 7, \mu > 7, or \mu < 7)\).

The researchers at the rural school conducted a simple random sample of n = 110 students on campus. They found that these students averaged 7.42 hours of sleep and the standard deviation of the amount of sleep for the students was 1.75 hours. A histogram of the sample is shown in Figure 4.14.

Before we can use a normal model for the sample mean or compute the standard error of the sample mean, we must verify conditions. (1) Because this is a simple random sample from less than 10% of the student body, the observations are independent. (2) The sample size in the sleep study is sufficiently large since it is greater than 30. (3) The data show moderate skew in Figure 4.14 and the presence of a couple of outliers. This skew and the outliers (which are not too extreme) are acceptable for a sample size of n = 110. With these conditions veri ed, the normal model can be safely applied to \(\bar {x}\) and the estimated standard error will be very accurate.

What is the standard deviation associated with \(\bar {x}\)? That is, estimate the standard error of \(\bar {x}\). 25

The hypothesis test will be evaluated using a significance level of \(\alpha = 0.05\). We want to consider the data under the scenario that the null hypothesis is true. In this case, the sample mean is from a distribution that is nearly normal and has mean 7 and standard deviation of about 0.17. Such a distribution is shown in Figure 4.15.

The shaded tail in Figure 4.15 represents the chance of observing such a large mean, conditional on the null hypothesis being true. That is, the shaded tail represents the p-value. We shade all means larger than our sample mean, \(\bar {x} = 7.42\), because they are more favorable to the alternative hypothesis than the observed mean.

We compute the p-value by finding the tail area of this normal distribution, which we learned to do in Section 3.1. First compute the Z score of the sample mean, \(\bar {x} = 7.42\):

\[Z = \dfrac {\bar {x} - \text {null value}}{SE_{\bar {x}}} = \dfrac {7.42 - 7}{0.17} = 2.47\]

Using the normal probability table, the lower unshaded area is found to be 0.993. Thus the shaded area is 1 - 0.993 = 0.007. If the null hypothesis is true, the probability of observing such a large sample mean for a sample of 110 students is only 0.007. That is, if the null hypothesis is true, we would not often see such a large mean.

We evaluate the hypotheses by comparing the p-value to the significance level. Because the p-value is less than the significance level \((p-value = 0.007 < 0.05 = \alpha)\), we reject the null hypothesis. What we observed is so unusual with respect to the null hypothesis that it casts serious doubt on H 0 and provides strong evidence favoring H A .

p-value as a tool in hypothesis testing

The p-value quantifies how strongly the data favor H A over H 0 . A small p-value (usually < 0.05) corresponds to sufficient evidence to reject H 0 in favor of H A .

TIP: It is useful to First draw a picture to find the p-value

It is useful to draw a picture of the distribution of \(\bar {x}\) as though H 0 was true (i.e. \(\mu\) equals the null value), and shade the region (or regions) of sample means that are at least as favorable to the alternative hypothesis. These shaded regions represent the p-value.

The ideas below review the process of evaluating hypothesis tests with p-values:

• The null hypothesis represents a skeptic's position or a position of no difference. We reject this position only if the evidence strongly favors H A .
• A small p-value means that if the null hypothesis is true, there is a low probability of seeing a point estimate at least as extreme as the one we saw. We interpret this as strong evidence in favor of the alternative.
• We reject the null hypothesis if the p-value is smaller than the significance level, \(\alpha\), which is usually 0.05. Otherwise, we fail to reject H 0 .
• We should always state the conclusion of the hypothesis test in plain language so non-statisticians can also understand the results.

The p-value is constructed in such a way that we can directly compare it to the significance level ( \(\alpha\)) to determine whether or not to reject H 0 . This method ensures that the Type 1 Error rate does not exceed the significance level standard.

If the null hypothesis is true, how often should the p-value be less than 0.05?

About 5% of the time. If the null hypothesis is true, then the data only has a 5% chance of being in the 5% of data most favorable to H A .

Exercise 4.31

Suppose we had used a significance level of 0.01 in the sleep study. Would the evidence have been strong enough to reject the null hypothesis? (The p-value was 0.007.) What if the significance level was \(\alpha = 0.001\)? 27

27 We reject the null hypothesis whenever p-value < \(\alpha\). Thus, we would still reject the null hypothesis if \(\alpha = 0.01\) but not if the significance level had been \(\alpha = 0.001\).

Exercise 4.32

Ebay might be interested in showing that buyers on its site tend to pay less than they would for the corresponding new item on Amazon. We'll research this topic for one particular product: a video game called Mario Kart for the Nintendo Wii. During early October 2009, Amazon sold this game for $46.99. Set up an appropriate (one-sided!) hypothesis test to check the claim that Ebay buyers pay less during auctions at this same time. 28

28 The skeptic would say the average is the same on Ebay, and we are interested in showing the average price is lower.

Exercise 4.33

During early October, 2009, 52 Ebay auctions were recorded for Mario Kart.29 The total prices for the auctions are presented using a histogram in Figure 4.17, and we may like to apply the normal model to the sample mean. Check the three conditions required for applying the normal model: (1) independence, (2) at least 30 observations, and (3) the data are not strongly skewed. 30

30 (1) The independence condition is unclear. We will make the assumption that the observations are independent, which we should report with any nal results. (2) The sample size is sufficiently large: \(n = 52 \ge 30\). (3) The data distribution is not strongly skewed; it is approximately symmetric.

H 0 : The average auction price on Ebay is equal to (or more than) the price on Amazon. We write only the equality in the statistical notation: \(\mu_{ebay} = 46.99\).

H A : The average price on Ebay is less than the price on Amazon, \(\mu _{ebay} < 46.99\).

29 These data were collected by OpenIntro staff.

Example 4.34

The average sale price of the 52 Ebay auctions for Wii Mario Kart was $44.17 with a standard deviation of $4.15. Does this provide sufficient evidence to reject the null hypothesis in Exercise 4.32? Use a significance level of \(\alpha = 0.01\).

The hypotheses were set up and the conditions were checked in Exercises 4.32 and 4.33. The next step is to find the standard error of the sample mean and produce a sketch to help find the p-value.

Because the alternative hypothesis says we are looking for a smaller mean, we shade the lower tail. We find this shaded area by using the Z score and normal probability table: \(Z = \dfrac {44.17 \times 46.99}{0.5755} = -4.90\), which has area less than 0.0002. The area is so small we cannot really see it on the picture. This lower tail area corresponds to the p-value.

Because the p-value is so small - specifically, smaller than = 0.01 - this provides sufficiently strong evidence to reject the null hypothesis in favor of the alternative. The data provide statistically signi cant evidence that the average price on Ebay is lower than Amazon's asking price.

Two-sided hypothesis testing with p-values

We now consider how to compute a p-value for a two-sided test. In one-sided tests, we shade the single tail in the direction of the alternative hypothesis. For example, when the alternative had the form \(\mu\) > 7, then the p-value was represented by the upper tail (Figure 4.16). When the alternative was \(\mu\) < 46.99, the p-value was the lower tail (Exercise 4.32). In a two-sided test, we shade two tails since evidence in either direction is favorable to H A .

Exercise 4.35 Earlier we talked about a research group investigating whether the students at their school slept longer than 7 hours each night. Let's consider a second group of researchers who want to evaluate whether the students at their college differ from the norm of 7 hours. Write the null and alternative hypotheses for this investigation. 31

Example 4.36 The second college randomly samples 72 students and nds a mean of \(\bar {x} = 6.83\) hours and a standard deviation of s = 1.8 hours. Does this provide strong evidence against H 0 in Exercise 4.35? Use a significance level of \(\alpha = 0.05\).

First, we must verify assumptions. (1) A simple random sample of less than 10% of the student body means the observations are independent. (2) The sample size is 72, which is greater than 30. (3) Based on the earlier distribution and what we already know about college student sleep habits, the distribution is probably not strongly skewed.

Next we can compute the standard error \((SE_{\bar {x}} = \dfrac {s}{\sqrt {n}} = 0.21)\) of the estimate and create a picture to represent the p-value, shown in Figure 4.18. Both tails are shaded.

31 Because the researchers are interested in any difference, they should use a two-sided setup: H 0 : \(\mu\) = 7, H A : \(\mu \ne 7.\)

An estimate of 7.17 or more provides at least as strong of evidence against the null hypothesis and in favor of the alternative as the observed estimate, \(\bar {x} = 6.83\).

We can calculate the tail areas by rst nding the lower tail corresponding to \(\bar {x}\):

\[Z = \dfrac {6.83 - 7.00}{0.21} = -0.81 \xrightarrow {table} \text {left tail} = 0.2090\]

Because the normal model is symmetric, the right tail will have the same area as the left tail. The p-value is found as the sum of the two shaded tails:

\[ \text {p-value} = \text {left tail} + \text {right tail} = 2 \times \text {(left tail)} = 0.4180\]

This p-value is relatively large (larger than \(\mu\)= 0.05), so we should not reject H 0 . That is, if H 0 is true, it would not be very unusual to see a sample mean this far from 7 hours simply due to sampling variation. Thus, we do not have sufficient evidence to conclude that the mean is different than 7 hours.

Example 4.37 It is never okay to change two-sided tests to one-sided tests after observing the data. In this example we explore the consequences of ignoring this advice. Using \(\alpha = 0.05\), we show that freely switching from two-sided tests to onesided tests will cause us to make twice as many Type 1 Errors as intended.

Suppose the sample mean was larger than the null value, \(\mu_0\) (e.g. \(\mu_0\) would represent 7 if H 0 : \(\mu\) = 7). Then if we can ip to a one-sided test, we would use H A : \(\mu > \mu_0\). Now if we obtain any observation with a Z score greater than 1.65, we would reject H 0 . If the null hypothesis is true, we incorrectly reject the null hypothesis about 5% of the time when the sample mean is above the null value, as shown in Figure 4.19.

Suppose the sample mean was smaller than the null value. Then if we change to a one-sided test, we would use H A : \(\mu < \mu_0\). If \(\bar {x}\) had a Z score smaller than -1.65, we would reject H 0 . If the null hypothesis is true, then we would observe such a case about 5% of the time.

By examining these two scenarios, we can determine that we will make a Type 1 Error 5% + 5% = 10% of the time if we are allowed to swap to the "best" one-sided test for the data. This is twice the error rate we prescribed with our significance level: \(\alpha = 0.05\) (!).

Caution: One-sided hypotheses are allowed only before seeing data

After observing data, it is tempting to turn a two-sided test into a one-sided test. Avoid this temptation. Hypotheses must be set up before observing the data. If they are not, the test must be two-sided.

Choosing a Significance Level

Choosing a significance level for a test is important in many contexts, and the traditional level is 0.05. However, it is often helpful to adjust the significance level based on the application. We may select a level that is smaller or larger than 0.05 depending on the consequences of any conclusions reached from the test.

• If making a Type 1 Error is dangerous or especially costly, we should choose a small significance level (e.g. 0.01). Under this scenario we want to be very cautious about rejecting the null hypothesis, so we demand very strong evidence favoring H A before we would reject H 0 .
• If a Type 2 Error is relatively more dangerous or much more costly than a Type 1 Error, then we should choose a higher significance level (e.g. 0.10). Here we want to be cautious about failing to reject H 0 when the null is actually false. We will discuss this particular case in greater detail in Section 4.6.

Significance levels should reflect consequences of errors

The significance level selected for a test should reflect the consequences associated with Type 1 and Type 2 Errors.

Example 4.38

A car manufacturer is considering a higher quality but more expensive supplier for window parts in its vehicles. They sample a number of parts from their current supplier and also parts from the new supplier. They decide that if the high quality parts will last more than 12% longer, it makes nancial sense to switch to this more expensive supplier. Is there good reason to modify the significance level in such a hypothesis test?

The null hypothesis is that the more expensive parts last no more than 12% longer while the alternative is that they do last more than 12% longer. This decision is just one of the many regular factors that have a marginal impact on the car and company. A significancelevel of 0.05 seems reasonable since neither a Type 1 or Type 2 error should be dangerous or (relatively) much more expensive.

Example 4.39

The same car manufacturer is considering a slightly more expensive supplier for parts related to safety, not windows. If the durability of these safety components is shown to be better than the current supplier, they will switch manufacturers. Is there good reason to modify the significance level in such an evaluation?

The null hypothesis would be that the suppliers' parts are equally reliable. Because safety is involved, the car company should be eager to switch to the slightly more expensive manufacturer (reject H 0 ) even if the evidence of increased safety is only moderately strong. A slightly larger significance level, such as \(\mu = 0.10\), might be appropriate.

Exercise 4.40

A part inside of a machine is very expensive to replace. However, the machine usually functions properly even if this part is broken, so the part is replaced only if we are extremely certain it is broken based on a series of measurements. Identify appropriate hypotheses for this test (in plain language) and suggest an appropriate significance level. 32

User Preferences

Content preview.

Arcu felis bibendum ut tristique et egestas quis:

• Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris
• Duis aute irure dolor in reprehenderit in voluptate
• Excepteur sint occaecat cupidatat non proident

Keyboard Shortcuts

6a.2 - steps for hypothesis tests, the logic of hypothesis testing section  .

A hypothesis, in statistics, is a statement about a population parameter, where this statement typically is represented by some specific numerical value. In testing a hypothesis, we use a method where we gather data in an effort to gather evidence about the hypothesis.

How do we decide whether to reject the null hypothesis?

• If the sample data are consistent with the null hypothesis, then we do not reject it.
• If the sample data are inconsistent with the null hypothesis, but consistent with the alternative, then we reject the null hypothesis and conclude that the alternative hypothesis is true.

Six Steps for Hypothesis Tests Section  

In hypothesis testing, there are certain steps one must follow. Below these are summarized into six such steps to conducting a test of a hypothesis.

• Set up the hypotheses and check conditions : Each hypothesis test includes two hypotheses about the population. One is the null hypothesis, notated as \(H_0 \), which is a statement of a particular parameter value. This hypothesis is assumed to be true until there is evidence to suggest otherwise. The second hypothesis is called the alternative, or research hypothesis, notated as \(H_a \). The alternative hypothesis is a statement of a range of alternative values in which the parameter may fall. One must also check that any conditions (assumptions) needed to run the test have been satisfied e.g. normality of data, independence, and number of success and failure outcomes.
• Decide on the significance level, \(\alpha \): This value is used as a probability cutoff for making decisions about the null hypothesis. This alpha value represents the probability we are willing to place on our test for making an incorrect decision in regards to rejecting the null hypothesis. The most common \(\alpha \) value is 0.05 or 5%. Other popular choices are 0.01 (1%) and 0.1 (10%).
• Calculate the test statistic: Gather sample data and calculate a test statistic where the sample statistic is compared to the parameter value. The test statistic is calculated under the assumption the null hypothesis is true and incorporates a measure of standard error and assumptions (conditions) related to the sampling distribution.
• Calculate probability value (p-value), or find the rejection region: A p-value is found by using the test statistic to calculate the probability of the sample data producing such a test statistic or one more extreme. The rejection region is found by using alpha to find a critical value; the rejection region is the area that is more extreme than the critical value. We discuss the p-value and rejection region in more detail in the next section.
• Make a decision about the null hypothesis: In this step, we decide to either reject the null hypothesis or decide to fail to reject the null hypothesis. Notice we do not make a decision where we will accept the null hypothesis.
• State an overall conclusion : Once we have found the p-value or rejection region, and made a statistical decision about the null hypothesis (i.e. we will reject the null or fail to reject the null), we then want to summarize our results into an overall conclusion for our test.

We will follow these six steps for the remainder of this Lesson. In the future Lessons, the steps will be followed but may not be explained explicitly.

Step 1 is a very important step to set up correctly. If your hypotheses are incorrect, your conclusion will be incorrect. In this next section, we practice with Step 1 for the one sample situations.

Hypothesis Testing

Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid.

A null hypothesis and an alternative hypothesis are set up before performing the hypothesis testing. This helps to arrive at a conclusion regarding the sample obtained from the population. In this article, we will learn more about hypothesis testing, its types, steps to perform the testing, and associated examples.

What is Hypothesis Testing in Statistics?

Hypothesis testing uses sample data from the population to draw useful conclusions regarding the population probability distribution . It tests an assumption made about the data using different types of hypothesis testing methodologies. The hypothesis testing results in either rejecting or not rejecting the null hypothesis.

Hypothesis Testing Definition

Hypothesis testing can be defined as a statistical tool that is used to identify if the results of an experiment are meaningful or not. It involves setting up a null hypothesis and an alternative hypothesis. These two hypotheses will always be mutually exclusive. This means that if the null hypothesis is true then the alternative hypothesis is false and vice versa. An example of hypothesis testing is setting up a test to check if a new medicine works on a disease in a more efficient manner.

Null Hypothesis

The null hypothesis is a concise mathematical statement that is used to indicate that there is no difference between two possibilities. In other words, there is no difference between certain characteristics of data. This hypothesis assumes that the outcomes of an experiment are based on chance alone. It is denoted as \(H_{0}\). Hypothesis testing is used to conclude if the null hypothesis can be rejected or not. Suppose an experiment is conducted to check if girls are shorter than boys at the age of 5. The null hypothesis will say that they are the same height.

Alternative Hypothesis

The alternative hypothesis is an alternative to the null hypothesis. It is used to show that the observations of an experiment are due to some real effect. It indicates that there is a statistical significance between two possible outcomes and can be denoted as \(H_{1}\) or \(H_{a}\). For the above-mentioned example, the alternative hypothesis would be that girls are shorter than boys at the age of 5.

Hypothesis Testing P Value

In hypothesis testing, the p value is used to indicate whether the results obtained after conducting a test are statistically significant or not. It also indicates the probability of making an error in rejecting or not rejecting the null hypothesis.This value is always a number between 0 and 1. The p value is compared to an alpha level, \(\alpha\) or significance level. The alpha level can be defined as the acceptable risk of incorrectly rejecting the null hypothesis. The alpha level is usually chosen between 1% to 5%.

Hypothesis Testing Critical region

All sets of values that lead to rejecting the null hypothesis lie in the critical region. Furthermore, the value that separates the critical region from the non-critical region is known as the critical value.

Hypothesis Testing Formula

Depending upon the type of data available and the size, different types of hypothesis testing are used to determine whether the null hypothesis can be rejected or not. The hypothesis testing formula for some important test statistics are given below:

• z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). \(\overline{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation and n is the size of the sample.
• t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\). s is the sample standard deviation.
• \(\chi ^{2} = \sum \frac{(O_{i}-E_{i})^{2}}{E_{i}}\). \(O_{i}\) is the observed value and \(E_{i}\) is the expected value.

We will learn more about these test statistics in the upcoming section.

Types of Hypothesis Testing

Selecting the correct test for performing hypothesis testing can be confusing. These tests are used to determine a test statistic on the basis of which the null hypothesis can either be rejected or not rejected. Some of the important tests used for hypothesis testing are given below.

Hypothesis Testing Z Test

A z test is a way of hypothesis testing that is used for a large sample size (n ≥ 30). It is used to determine whether there is a difference between the population mean and the sample mean when the population standard deviation is known. It can also be used to compare the mean of two samples. It is used to compute the z test statistic. The formulas are given as follows:

• One sample: z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).
• Two samples: z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing t Test

The t test is another method of hypothesis testing that is used for a small sample size (n < 30). It is also used to compare the sample mean and population mean. However, the population standard deviation is not known. Instead, the sample standard deviation is known. The mean of two samples can also be compared using the t test.

• One sample: t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\).
• Two samples: t = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing Chi Square

The Chi square test is a hypothesis testing method that is used to check whether the variables in a population are independent or not. It is used when the test statistic is chi-squared distributed.

One Tailed Hypothesis Testing

One tailed hypothesis testing is done when the rejection region is only in one direction. It can also be known as directional hypothesis testing because the effects can be tested in one direction only. This type of testing is further classified into the right tailed test and left tailed test.

Right Tailed Hypothesis Testing

The right tail test is also known as the upper tail test. This test is used to check whether the population parameter is greater than some value. The null and alternative hypotheses for this test are given as follows:

\(H_{0}\): The population parameter is ≤ some value

\(H_{1}\): The population parameter is > some value.

If the test statistic has a greater value than the critical value then the null hypothesis is rejected

Left Tailed Hypothesis Testing

The left tail test is also known as the lower tail test. It is used to check whether the population parameter is less than some value. The hypotheses for this hypothesis testing can be written as follows:

\(H_{0}\): The population parameter is ≥ some value

\(H_{1}\): The population parameter is < some value.

The null hypothesis is rejected if the test statistic has a value lesser than the critical value.

Two Tailed Hypothesis Testing

In this hypothesis testing method, the critical region lies on both sides of the sampling distribution. It is also known as a non - directional hypothesis testing method. The two-tailed test is used when it needs to be determined if the population parameter is assumed to be different than some value. The hypotheses can be set up as follows:

\(H_{0}\): the population parameter = some value

\(H_{1}\): the population parameter ≠ some value

The null hypothesis is rejected if the test statistic has a value that is not equal to the critical value.

Hypothesis Testing Steps

Hypothesis testing can be easily performed in five simple steps. The most important step is to correctly set up the hypotheses and identify the right method for hypothesis testing. The basic steps to perform hypothesis testing are as follows:

• Step 1: Set up the null hypothesis by correctly identifying whether it is the left-tailed, right-tailed, or two-tailed hypothesis testing.
• Step 2: Set up the alternative hypothesis.
• Step 3: Choose the correct significance level, \(\alpha\), and find the critical value.
• Step 4: Calculate the correct test statistic (z, t or \(\chi\)) and p-value.
• Step 5: Compare the test statistic with the critical value or compare the p-value with \(\alpha\) to arrive at a conclusion. In other words, decide if the null hypothesis is to be rejected or not.

Hypothesis Testing Example

The best way to solve a problem on hypothesis testing is by applying the 5 steps mentioned in the previous section. Suppose a researcher claims that the mean average weight of men is greater than 100kgs with a standard deviation of 15kgs. 30 men are chosen with an average weight of 112.5 Kgs. Using hypothesis testing, check if there is enough evidence to support the researcher's claim. The confidence interval is given as 95%.

Step 1: This is an example of a right-tailed test. Set up the null hypothesis as \(H_{0}\): \(\mu\) = 100.

Step 2: The alternative hypothesis is given by \(H_{1}\): \(\mu\) > 100.

Step 3: As this is a one-tailed test, \(\alpha\) = 100% - 95% = 5%. This can be used to determine the critical value.

1 - \(\alpha\) = 1 - 0.05 = 0.95

0.95 gives the required area under the curve. Now using a normal distribution table, the area 0.95 is at z = 1.645. A similar process can be followed for a t-test. The only additional requirement is to calculate the degrees of freedom given by n - 1.

Step 4: Calculate the z test statistic. This is because the sample size is 30. Furthermore, the sample and population means are known along with the standard deviation.

z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).

\(\mu\) = 100, \(\overline{x}\) = 112.5, n = 30, \(\sigma\) = 15

z = \(\frac{112.5-100}{\frac{15}{\sqrt{30}}}\) = 4.56

Step 5: Conclusion. As 4.56 > 1.645 thus, the null hypothesis can be rejected.

Hypothesis Testing and Confidence Intervals

Confidence intervals form an important part of hypothesis testing. This is because the alpha level can be determined from a given confidence interval. Suppose a confidence interval is given as 95%. Subtract the confidence interval from 100%. This gives 100 - 95 = 5% or 0.05. This is the alpha value of a one-tailed hypothesis testing. To obtain the alpha value for a two-tailed hypothesis testing, divide this value by 2. This gives 0.05 / 2 = 0.025.

Related Articles:

• Probability and Statistics
• Data Handling

Important Notes on Hypothesis Testing

• Hypothesis testing is a technique that is used to verify whether the results of an experiment are statistically significant.
• It involves the setting up of a null hypothesis and an alternate hypothesis.
• There are three types of tests that can be conducted under hypothesis testing - z test, t test, and chi square test.
• Hypothesis testing can be classified as right tail, left tail, and two tail tests.

Examples on Hypothesis Testing

• Example 1: The average weight of a dumbbell in a gym is 90lbs. However, a physical trainer believes that the average weight might be higher. A random sample of 5 dumbbells with an average weight of 110lbs and a standard deviation of 18lbs. Using hypothesis testing check if the physical trainer's claim can be supported for a 95% confidence level. Solution: As the sample size is lesser than 30, the t-test is used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) > 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 5, s = 18. \(\alpha\) = 0.05 Using the t-distribution table, the critical value is 2.132 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = 2.484 As 2.484 > 2.132, the null hypothesis is rejected. Answer: The average weight of the dumbbells may be greater than 90lbs
• Example 2: The average score on a test is 80 with a standard deviation of 10. With a new teaching curriculum introduced it is believed that this score will change. On random testing, the score of 38 students, the mean was found to be 88. With a 0.05 significance level, is there any evidence to support this claim? Solution: This is an example of two-tail hypothesis testing. The z test will be used. \(H_{0}\): \(\mu\) = 80, \(H_{1}\): \(\mu\) ≠ 80 \(\overline{x}\) = 88, \(\mu\) = 80, n = 36, \(\sigma\) = 10. \(\alpha\) = 0.05 / 2 = 0.025 The critical value using the normal distribution table is 1.96 z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) z = \(\frac{88-80}{\frac{10}{\sqrt{36}}}\) = 4.8 As 4.8 > 1.96, the null hypothesis is rejected. Answer: There is a difference in the scores after the new curriculum was introduced.
• Example 3: The average score of a class is 90. However, a teacher believes that the average score might be lower. The scores of 6 students were randomly measured. The mean was 82 with a standard deviation of 18. With a 0.05 significance level use hypothesis testing to check if this claim is true. Solution: The t test will be used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) < 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 6, s = 18 The critical value from the t table is -2.015 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = \(\frac{82-90}{\frac{18}{\sqrt{6}}}\) t = -1.088 As -1.088 > -2.015, we fail to reject the null hypothesis. Answer: There is not enough evidence to support the claim.

go to slide go to slide go to slide

Book a Free Trial Class

FAQs on Hypothesis Testing

What is hypothesis testing.

Hypothesis testing in statistics is a tool that is used to make inferences about the population data. It is also used to check if the results of an experiment are valid.

What is the z Test in Hypothesis Testing?

The z test in hypothesis testing is used to find the z test statistic for normally distributed data . The z test is used when the standard deviation of the population is known and the sample size is greater than or equal to 30.

What is the t Test in Hypothesis Testing?

The t test in hypothesis testing is used when the data follows a student t distribution . It is used when the sample size is less than 30 and standard deviation of the population is not known.

What is the formula for z test in Hypothesis Testing?

The formula for a one sample z test in hypothesis testing is z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) and for two samples is z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

What is the p Value in Hypothesis Testing?

The p value helps to determine if the test results are statistically significant or not. In hypothesis testing, the null hypothesis can either be rejected or not rejected based on the comparison between the p value and the alpha level.

What is One Tail Hypothesis Testing?

When the rejection region is only on one side of the distribution curve then it is known as one tail hypothesis testing. The right tail test and the left tail test are two types of directional hypothesis testing.

What is the Alpha Level in Two Tail Hypothesis Testing?

To get the alpha level in a two tail hypothesis testing divide \(\alpha\) by 2. This is done as there are two rejection regions in the curve.

Sydney dominate Carlton as Swans tighten grip on top of AFL ladder in SCG Marn Grook match

Sport Sydney dominate Carlton as Swans tighten grip on top of AFL ladder in SCG Marn Grook match

Sydney have soared two wins clear on top of the AFL ladder but were left with several concerns after crushing Carlton by 52 points at a heaving SCG.

The Blues exploded out of the blocks on Friday night to lead 27-1 midway through the opening term.

But it was all pain from that point on as Sydney piled on 14 of the next 16 goals to soar to the 17.15 (117) to 9.11 (65) victory.

The third quarter was particularly brutal as Sydney dominated the centre clearances 7-1 on the way to posting seven goals to two.

The result lifted Sydney two wins clear of second-placed Geelong, and marks the first time since 1945 the club has won nine of its first 10 matches.

Even if third-placed Essendon beat North Melbourne as expected on Sunday, the Swans will still be one and a half wins clear on top.

But the Swans were left with several concerns.

Star midfielder Isaac Heeney, who kicked three goals from 24 disposals in another hot display, was left hobbling after injuring his ankle late in the match.

Robbie Fox was subbed out in the third term after appearing to dislocate his right shoulder when he was bowled over by a sliding Jordan Boyd.

And the Swans will also be sweating on what the match review officer makes of Chad Warner's high fend-off that struck Carlton ruckman Marc Pittonet flush in the face with a forearm/elbow.

Warner was a standout with three goals and 28 disposals.

Nic Newman racked up 32 disposals for Carlton, but Sam Walsh (20 possessions) struggled for influence after being tagged by James Jordon.

Star Blues defender Jacob Weitering was subbed out at half-time after struggling through a corked quad he suffered in the opening term.

Look back at how the action unfolded in our live blog.

• 12h ago 12 hours ago Fri 17 May 2024 at 12:35pm Chad Warner wins the Goodes-O'Loughlin Medal
• 12h ago 12 hours ago Fri 17 May 2024 at 12:19pm FT: Sensational Sydney crush Carlton by 52 points
• 12h ago 12 hours ago Fri 17 May 2024 at 11:43am 3QT: It's all Sydney as the Swans bust out to a 45-point lead

To leave a comment on the blog, please log in or sign up for an ABC account.

Live updates

Sydney vs carlton.

By Dean Bilton

That's it from the blog tonight

Sydney fans will sleep soundly tonight and can enjoy the rest of this weekend safe in the knowledge that their gap at the top has grown a little bit. The Swans are the real deal, and proved it yet again tonight.

That's it from me tonight! The blog will be back tomorrow as Sir Doug Nicholls Round continues, and I hope you will join us then. Goodnight!

Chad Warner wins the Goodes-O'Loughlin Medal

No surprises there, despite some other stand outs. Warner was pretty breathtaking tonight, especially when the game was there to be won in the second quarter.

Warner finishes up with 28 disposals, three goals, 14 score involvements and 561 metres gained. And a whole lot of sidesteps.

FT: Sensational Sydney crush Carlton by 52 points

How do you stop them? I guess you could ask Richmond, who remarkably are the only team to beat Sydney this year.

But when they are really cooking, and their engine room is playing as well as this, it's hard to see many teams coming close. This was a clinical win, built from a resolute base of team defence and elite pressure and elevated by incredible individuals like Chad Warner and Isaac Heeney.

Carlton might have reached a tipping point with their injury list, and certainly without Jacob Weitering in the second half they looked very light on down back. The Blues looked tired and unable to match the Swans physically, which probably is to be expected.

The Swans are now two games clear on top of the ladder, and by most metrics the best team in the league right now. There is a long way to go before we get to September and the games that really matter but right here, right now, Sydney are the best team in it by some distance.

Into the final minutes of this one

We're into the part of the season where fatigue is starting to hit every team, which explains the fairly pedestrian pace this dead rubber of a last quarter has been played at.

Sydney are a couple of minutes away from a convincing win.

Hayden McLean with the accidental goal! Everything's coming up Sydney

McLean was trying to chip a little pass to Joel Amartey in the goalsquare there, but overhit his kick.

No matter, the ball easily cleared Amartey and his defender and instead trickled through for another goal. Been that sort of night for Carlton.

James Rowbottom won't be denied! Another Sydney goal

Chad Warner broke a couple of tackles and dished off to Rowbottom who broke a couple more. He threw the ball on his boot and watched it tumble end over end and straight through the middle.

Sydney's midfield has had a day out.

George Hewett kicks Carlton's ninth for the night

So the heat is already well and truly out of this one, but the margin is now under 40 points. I reckon we'll see some goals at both ends in this quarter, not unlike last night's bizarre shootout in Darwin.

Better pressure from the Blues inside 50 forced the turnover and Hewett marked the squaring kick directly in front of goal.

Isaac Heeney has three and the Swans hit straight back

Heeney and Warner seem locked in a little battle tonight, with Brownlow votes up for grabs and goals seemingly easy to come by.

Both now have three goals, the latest for Heeney coming from a 40 metre set shot from an angle.

Zac Williams kicks a goal for Carlton

Nick Blakey got a little bit excited and tried to take on the whole Carlton defence with one of his lizardy runs, but Williams wasn't having it.

His tackle was excellent, and the set shot from the resulting free kick just carried the line.

3QT: It's all Sydney as the Swans bust out to a 45-point lead

Not the contest we had hoped for, or thought we were getting during the first quarter. Instead Sydney have completely overwhelmed Carlton, who have run out of options and a little bit of fight in that third quarter.

There's just no stopping Chad Warner and Isaac Heeney right now, but every single Swan out there is playing his role to perfection and at 100 per cent intensity at all times. They've suffocated Carlton to breaking point, and now all there is to determine is the final margin.

Sydney will soon be two games clear on top of the ladder, Carlton could soon be out of the top eight.

Matt Kennedy pulls one back for Carlton

Just moments after Charlie Curnow hit the post from a long-range set shot, Kennedy converts from much closer range.

Muted celebrations as the third quarter winds down.

Hayden McLean takes the Sydney lead out to 52 points

Remember Carlton kicked the first four goals of this game. It has been an absolute hammering since about halfway through the first quarter.

So many goals are coming from straight from centre clearance, and the same is true there. Heeney laces out McLean, McLean finishes it off.

Harry Cunningham finishes off another slick Sydney move

End to end in the blink of an eye, Sydney have cut Carlton to ribbons after quarter-time tonight.

Cunningham was the final link in a sumptuous handball chain from half-back, and having run about 100 metres to get the footy he wasn't going to waste his moment. Great goal.

Chad Warner has three! What a performance!

At some point, Michael Voss might want to think about putting a Carlton player somewhere in the vicinity of Chad Warner.

He was just lurking about 15 metres off the pack inside 50, all by himself. Eventually the handball came out to him, and he's just not missing tonight. Sydney running away with it now.

Sam Wicks kicks another Sydney goal! It's an avalanche!

It's a bloodbath in the centre clearances now. The Swans are just walking straight out the front door and raffling off goals.

Adams, Grundy and Wicks combined that time with slick hands. The latter nailed the goal, and half the team got together to celebrate.

Isaac Heeney has two and it's all Sydney now

This is kinda what we feared might happen in this third quarter, as Sydney look just about ready to blow Carlton off the park now.

James Jordon's set up kick was great, but Heeney is just too strong for any Blues defender when he goes forward. He marks and goals from close range.

Chad Warner bursts out of the middle and kicks another!

Pristine Sydney football from the centre clearance! Grundy to Rowbottom, Rowbottom to Warner, Warner to goal.

Amartey got pretty handsy with the Blues defender on the line to usher the ball through, but the umpire was happy enough. Warner was even happier.

Isaac Heeney kicks his first of the night

With Weitering now out of action, that Carlton defence is going to be stretched all night. The Swans did well to bring it to ground, and McDonald's little tap gave Heeney the chance to snap another one through.

Curnow's early goal is quickly cancelled out.

Jacob Weitering has been subbed out

He copped a cork in the first quarter and clearly has been hampered by it, but at half-time the Blues have pulled the trigger on making that sub.

Carlton can't take a trick on the injury front.

• Sports Betting
• Sports Entertainment
• Transactions

Recommended

Igor shesterkin’s stone-cold third-period stop preserved the rangers’ wild rally.

• View Author Archive
• Email the Author
• Follow on Twitter
• Get author RSS feed

Contact The Author

Thanks for contacting us. We've received your submission.

Thanks for contacting us. We've received your submission.

Chris Kreider’s natural hat trick put the Rangers in front.

Igor Shesterkin made sure they stayed there.

The Russian netminder turned in perhaps the biggest of his 33 saves on the night with 2:43 left in the third period to preserve a 4-3 Blueshirts lead en route to a 5-3 win over the Hurricanes in Game 6 that sent the Presidents’ Trophy winners into the Eastern Conference Final.

Carolina forward Andrei Svechnikov found himself alone with the puck in the slot after a faceoff in the Rangers’ end, with only Shesterkin between him and a tie game in Raleigh, N.C.

The Hurricanes winger — who had already racked up a pair of assists on the evening — tagged a snap shot for the far post, but Shesterkin coolly flashed his blocker and directed the puck safely into the corner.

The save was emblematic of the goalie’s even-keeled nature, with his demeanor not swaying despite giving up three goals before the halfway point of the game.

“There’s an edge to him if we’re losing, but he’s a pretty consistent level-headed guy all the time,” Jacob Trouba told The Post’s Larry Brooks on Wednesday . “But this morning, you could tell he was gearing up. He knows what’s on the line.”

Back-to-back losses after three straight wins to open the semifinal series had the Rangers feeling the pressure — especially after falling flat in a 4-1 Game 5 loss that saw the Hurricanes post four third-period goals, three of them getting behind Shesterkin .

Follow The Post’s coverage of the Rangers in the NHL playoffs

• Chris Kreider hat trick completes Rangers insane rally to eliminate Hurricanes
• Brooks: Chris Kreider adds another signature playoff moment to storied Rangers career
• Rangers’ status as NHL’s comeback kings furthered in wild Game 6 win
• Igor Shesterkin’s stone-cold third-period stop preserved the Rangers’ wild rally

Carolina wasn’t going to replicate that kind of luck on Thursday night.

Not against the Rangers, who led the NHL with 28 come-from-behind wins during the regular season, and not against Shesterkin, who backstopped the team to eight playoff wins so far this year.

Igor Shesterkin DENIES Andrei Svechnikov to preserve the lead 😮‍💨 pic.twitter.com/ZHLbQtKPCL — Gino Hard (@GinoHard_) May 17, 2024

He’ll have a chance for more with the Rangers headed to the Eastern Conference Final, awaiting the winner of the Bruins-Panthers series.

Share this article:

• Mattress Toppers
• Sheets & Bedding
• Sleep Products
• Cleaning & Laundry
• Heating & Cooling
• Home Office
• Kitchen & Dining
• Storage & Organization
• Wine & Bar
• Accessories
• Handbags & Purses
• Lingerie & Sleepwear
• Outdoor & Fitness Apparel
• Kids Clothes & Accessories
• Pregnancy & Postpartum
• Toys & Books
• Apparel & Accessories
• Camping & Hiking
• Fishing & Hunting
• Tennis & Racket Sports
• Tools & Tech
• Training & Recovery
• Water Sports
• Winter Sports
• Personal Products
• Wellness & Self Care
• Computers, Tablets & Accessories
• Online Tools
• Smart Home Devices
• Smartphones, Smartwatches & Accessories
• Hotels & Lodging
• Travel Products
• Father's Day
• Mother's Day
• Valentine's Day
• Amazon Prime Day
• Beauty & Grooming
• Tech & Electronics
• Travel Deals
• Mattress & Sleep

How To Choose A Bike: A Comprehensive Buyer’s Guide For Every Cyclist

• Share to Facebook
• Share to Twitter
• Share to Linkedin

If you’re in the market for a new bike, the hundreds, if not thousands, of options out there can make the journey feel overwhelming. Fortunately, our gear experts have bought a variety of bikes over the years, which has helped us put together a step-by-step guide on how to choose a bike. From demystifying frame sizes to the game-changing benefits of a professional fitting, we've packed it all in to ensure the bike you purchase is the right one for you.

Follow our comprehensive guide to choose the right bike that meets your needs.

The Best Memorial Day Mattress Sales, According To Our Deals Editors

The 10 best summer fragrances for men, from woody to citrus scents, step 1: consider your needs.

Start by thinking about how you intend to use your bike. Are you commuting to work, riding around town for fun, tackling trails or spending hours on the road? Understanding what purpose your bike will serve allows you to narrow down your options.

Road, Gravel, Mountain And Hybrid Bikes

Road, gravel, mountain and hybrid bikes are the predominant types you’re likely to encounter, and each have their own personalities that are perfect for different kinds of adventures.

• Road bikes : These are the speed demons of the bunch, zipping along smooth pavement with their sleek frames and skinny tires. True “road” bikes often come in a drop bar style that’s more aerodynamic and offers varying levels of comfort based on the bike. Don’t feel obligated to buy a drop bar bike if it doesn’t feel right—flat bar bikes (the more traditional approach) are perfectly fine as well.
• Gravel bikes : These are like the road bike’s adventurous cousin that’s ready to tackle anything from country lanes to off-road trails with their beefed-up tires and sturdy builds. These days, the trend is for gravel bikes to clear tires up to 50mm, which is more than most people need for a commute, but it gives you the flexibility to have a second set of wheels on hand for multiple kinds of riding without the need for another bike.
• Mountain bikes : As you can probably guess, these are built tough with suspension systems and knobby tires to conquer rocky trails and steep terrain. Different types of mountain bikes (enduro, hardtail, cross-country and so on) cater to different off-road pursuits.
• Hybrid bikes: These offer a bit of everything, with comfy seats, wider tires for stability and just enough toughness to handle a bit of off-roading while still cruising smoothly on the streets.

Different types of bikes cater to different scenarios.

Step 2: Set A Budget

As tempting as it may be to daydream over a pricey bike, figure out how much you’re willing to spend on a new ride before starting your search. Specific types of bikes with similar components will often fall into a specific price range, which makes it easier to determine how much you should expect to pay.

Keep in mind that higher-end components and lightweight frame materials like carbon are more expensive—the key is to find a bike that strikes a balance between price and capability. And while you’re putting a budget together, don't forget to factor in the cost of gear, like a helmet , shoes and even a hitch rack , if you don’t have the essentials.

Bike Frame Materials

There are three main frame materials to choose from: carbon fiber, titanium and aluminum/steel.

• Carbon fiber: This is the most popular choice for higher-end bikes—you can find full carbon frames in the sub-$3,000 price category. It’s a flexible material with a comfortable ride quality and the engineering allows it to mold into just about any shape, which is why bike manufacturers like it. One of the not-so-secret secrets of mid-range to higher-end carbon bike frames is that unless they’re made in Europe or America, they most likely come from the same handful of factories in Taiwan.
• Titanium: This material is popular among enthusiasts for a stiffer ride and bombproof durability.“It does a remarkable job absorbing high-frequency vibrations and low-frequency impacts from things like potholes,” says Moots president Nate Bradley. Titanium is typically more expensive than carbon, but it’s built for life. Bradley adds that metals can bend to absorb impact, whereas carbon is ultimately trying to resist bending until it delaminates, cracks or fails “catastrophically.” It is much more complicated to repair a carbon frame compared to titanium or steel, although both processes are expensive.
• Aluminum And Steel: It’s getting harder to find a quality aluminum bike, but “steel is real” for commuters and certain road bikes. Aluminum and steel frames are cheaper, but they still maintain high durability and easy repairability compared to their carbon counterparts.

Carbon fiber bikes offer a lightweight, compliant ride for hours of comfort in the saddle.

Step 3: Determine The Right Size

Getting the right size bike is key to long-term comfort and performance. If you're not sure what size you need, pop into a local bike shop, where a team can help guide you to the right bike that fits your dimensions. Plus, it’s never a bad idea to start a healthy relationship with a local shop for maintenance and part needs if you don’t plan on tackling repairs at home.

If you intend to buy from a direct-to-consumer (DTC) brand that doesn’t work with local bike shops, refer to the brand’s online size chart that often uses your height and inner leg length to recommend a size; from our experience, these tend to be accurate.

Step 4: Test Ride

Once you have a short list of bikes that meet your needs and fall within your budget, it’s time to take each of them for a test ride. Like visiting a car dealership to test drive a new vehicle, test riding a bike helps you understand if it’s comfortable, and it’s a great opportunity to familiarize yourself with a bike’s features.

This is also a great time to discuss sizing if you’re still unsure of what size bike to get, which might be the case if you fall between sizes. Share any noticeable discomforts so the experts can make minor adjustments to the seat, handlebars and head tube—these changes can drastically improve comfort.

Bicycle Components

If you’ve ever seen a luxury car commercial, you know the manufacturer is advertising their top-of-the-line car, and in a similar way, the bike industry has a habit of putting the highest-end (and most expensive) bikes on its advertising. While these bikes look great, the vast majority of riders don’t necessarily need that level of performance.

For newer or mid-range riders, here’s a short list of some componentry to look for that strikes a balance between price and capability:

• Shimano 105 : This is an excellent drivetrain choice that’s slightly higher on Shimano’s model list, but it’s a quality, easily-serviceable mid-range choice with dependable parts (such as chainrings, chains and derailleurs). Shimano also just released the Essa system as an affordable, commuter-focused option.
• SRAM Apex And Rival : While Rival is the American company’s legacy mid-range system, Apex is its newest, and perhaps best value. Rival is still available in a mechanical set while Apex is only electronic (you’ll see it named “AXS”). Apex is typically reserved for gravel bikes while Rival is for road and gravel.
• Campagnolo : Known as “Campy” in some corners of the bike world, this is a legacy Italian brand known for similar ranges as SRAM and Shimano, but more for a specific kind of bike enthusiast. The brand isn’t as popular in the U.S. as it once was, but it remains a very good option for certain riders. There is absolutely nothing wrong with a Campy drivetrain, but it can be harder to find replacement parts.
• MicroShift : Unless you’re really hunting for a bargain, it’s probably best to avoid MicroShift. The brand is getting better in terms of quality, but for similar pricing, you should be able to find a much more reliable, gently-used option from Shimano or SRAM.

Different componentry will influence your bike's abilities on and off the beaten path.

Step 5: Schedule A Bike Fitting

As easy as it may be to hop on a bike and ride off into the sunset, we cannot stress enough the importance a professional bike fitting, especially if you’re investing in a performance-oriented bike. A professional fitter will assess your goals and your experience on a bike as well as your dimensions to make small adjustments that improve comfort and pedaling efficiency.

Additional Buying Advice

Because the bike industry exploded as a result of the COVID-19 pandemic, there are more used bikes on the secondhand market than ever before. If you don’t mind a bike from a couple of seasons ago, you can often find a really great deal on a really great bike. Stick with a reputable site like Pink Bike or The Pro’s Closet , or head to a shop that backs up their used bikes with quality guarantees.

Shop DTC To Spend Less

A growing number of the best bike brands sell direct to the consumer, or DTC, as a means of sidestepping a relationship with a local bike shop to save you money, and these are a great option if you’ve done your research on a brand you like. Even big brands like Specialized are moving online and away from the dealer model.

What About Electric Bikes?

Stuart Sundell-Norlin, associate category merchandise manager at Christy Sports , specializes in helping riders find sensible e-bikes they’ll actually use.

“There are more options than ever are out there to get yourself into an e-bike with great sale pricing both at the dealer and vendor levels,” he says, “with no shortage of price points.”

There are power-assisted models available for just about every need. Popular cargo bikes often have a hub-drive motor located in the rear wheel while commuter- and performance-oriented electric bikes have the battery in the middle of the frame. Higher-spec bikes will have that battery built in, which is better for performance, but also more difficult to service.

Sundell-Norlin notes that if you have any interest in using an e-bike off-road, a mountain-specific model could be a great choice. There are also a growing number of “e-gravel” options that mirror their non-electric siblings.

Much like buying an electric vehicle, don’t get caught up in the formal range ratings for an e-bike. “Range will vary with rider weight, usage habits and terrain,” Sundell-Norlin explains. “A fully-loaded bike with groceries will certainly have less range than its empty counterpart.”

How Much Should A Beginner Bike Cost?

For beginners, considering a mid-level bike in the price range of $1,000 to $1,500 is a smart move. These bikes offer a step up in quality with reliable Shimano or SRAM components, known for their smooth performance and durability. By investing a bit more, you'll enjoy smoother shifting, better braking and overall improved ride quality compared to cheaper options. Plus, mid-level bikes often come with lighter frames and more comfortable features, making your experience more enjoyable, especially on longer rides.

Other Bike Stories To Shop

• Best Bike Pumps
• Best Bike Lights
• Best Bike Locks
• Best Bike Storage Solutions

• Editorial Standards
• Reprints & Permissions

A .gov website belongs to an official government organization in the United States.

A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

• Risk Factors
• Providing Care
• Living with Diabetes
• Clinical Guidance
• DSMES for Health Care Providers
• Prevent Type 2 Diabetes: Talking to Your Patients About Lifestyle Change
• Employers and Insurers
• Community-based Organizations (CBOs)
• Toolkits for Diabetes Educators and Community Health Workers
• National Diabetes Statistics Report
• Reports and Publications
• Current Research Projects
• National Diabetes Prevention Program
• State, Local, and National Partner Diabetes Programs for Public Health
• Diabetes Self-Management Education and Support (DSMES) Toolkit

Testing for Diabetes and Prediabetes: A1C

• The A1C test—also known as the hemoglobin A1C or HbA1c test—is a simple blood test.
• Your A1C is used to diagnose prediabetes and diabetes, and monitor your progress.
• Find out more about the test, and your A1C goals.

What does the A1C test measure?

When you check your blood sugar at home, it's a snapshot of a single point in time. But your blood sugar constantly changes, so this doesn't give you a complete picture. The A1C test measures your average blood sugar levels over the past 3 months.

When sugar enters your bloodstream, it attaches to hemoglobin, a protein in your red blood cells. Everybody has some sugar attached to their hemoglobin, but people with higher blood sugar levels have more. The A1C test measures the percentage of your red blood cells that have sugar-coated hemoglobin. Your red blood cells regenerate roughly every 3 months. That's why the A1C test measures your blood sugar levels from that time period.

A1C is just part of your toolkit‎

Getting tested.

The A1C test is done in a doctor's office or at a lab. You will have a blood sample drawn from your finger or arm. You don't need to fast before an A1C test, but your doctor may run other tests like cholesterol at the same time that might require fasting.

Get an A1C test if you're over age 45. A1C testing is also recommended if you're younger, have overweight, and any other risk factor for prediabetes and type 2 diabetes .

If your result is normal:

Your doctor will recommend a re-testing schedule based on your age and risk factors.

If your result shows you have prediabetes:

There's good news. You can take steps right away to reverse prediabetes or to prevent or delay type 2 diabetes. Talk to your doctor about how to get started. They'll likely recommend you repeat your A1C every 1 to 2 years.

If your result shows you have diabetes:

Your doctor will give you instructions how to manage your condition with lifestyle changes, and may prescribe you medicine. Most people with diabetes have their A1C tested at least twice a year. You may need to check more often based on your management plan or other health conditions. Ask your doctor how often is right for you.

A1C results

The following ranges are used to diagnose prediabetes and diabetes:

• Normal: below 5.7%
• Prediabetes: 5.7% to 6.4%
• Diabetes: 6.5% or above

When living with diabetes, your A1C also shows how well managed your condition is. Your A1C can estimate your average blood sugar (although it may not account for any spikes or lows):

Estimated average glucose mg/dL

For most people with diabetes, the A1C goal is 7% or less . Your doctor will determine your specific goal based on your full medical history. Higher A1C levels are linked to health complications, so reaching and maintaining your goal is key to living well with diabetes.

With lifestyle changes and medicines (if prescribed), some people with diabetes can reach an A1C below 6.5%. This doesn't mean that their condition went away, but it does usually mean their blood sugar is well managed.

Things that affect A1C accuracy

Several factors can falsely increase or decrease your A1C result, including:

• Severe anemia.
• Kidney failure.
• Liver disease.
• Certain blood disorders like sickle cell anemia or thalassemia.
• Certain medicines, including opioids and some HIV medications.
• Blood loss or blood transfusions.
• Early or late pregnancy.

Let your doctor know if any of these factors apply to you.

Diabetes is a chronic disease that affects how your body turns food into energy. About 1 in 10 Americans has diabetes.

For Everyone

Health care providers, public health.