critical value approach in hypothesis testing

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Statistics By Jim

Making statistics intuitive

Critical Value: Definition, Finding & Calculator

By Jim Frost 2 Comments

What is a Critical Value?

A critical value defines regions in the sampling distribution of a test statistic. These values play a role in both hypothesis tests and confidence intervals. In hypothesis tests, critical values determine whether the results are statistically significant. For confidence intervals, they help calculate the upper and lower limits.

In both cases, critical values account for uncertainty in sample data you’re using to make inferences about a population . They answer the following questions:

• How different does the sample estimate need to be from the null hypothesis to be statistically significant?
• What is the margin of error (confidence interval) around the sample estimate of the population parameter ?

In this post, I’ll show you how to find critical values, use them to determine statistical significance, and use them to construct confidence intervals. I also include a critical value calculator at the end of this article so you can apply what you learn.

Because most people start learning with the z-test and its test statistic, the z-score, I’ll use them for the examples throughout this post. However, I provide links with detailed information for other types of tests and sampling distributions.

Related posts : Sampling Distributions and Test Statistics

Using a Critical Value to Determine Statistical Significance

In this context, the sampling distribution of a test statistic defines the probability for ranges of values. The significance level (α) specifies the probability that corresponds with the critical value within the distribution. Let’s work through an example for a z-test.

The z-test uses the z test statistic. For this test, the z-distribution finds probabilities for ranges of z-scores under the assumption that the null hypothesis is true. For a z-test, the null z-score is zero, which is at the central peak of the sampling distribution. This sampling distribution centers on the null hypothesis value, and the critical values mark the minimum distance from the null hypothesis required for statistical significance.

Critical values depend on your significance level and whether you’re performing a one- or two-sided hypothesis. For these examples, I’ll use a significance level of 0.05. This value defines how improbable the test statistic must be to be significant.

Related posts : Significance Levels and P-values and Z-scores

Two-Sided Tests

Two-sided hypothesis tests have two rejection regions. Consequently, you’ll need two critical values that define them. Because there are two rejection regions, we must split our significance level in half. Each rejection region has a probability of α / 2, making the total likelihood for both areas equal the significance level.

The probability plot below displays the critical values and the rejection regions for a two-sided z-test with a significance level of 0.05. When the z-score is ≤ -1.96 or ≥ 1.96, it exceeds the cutoff, and your results are statistically significant.

One-Sided Tests

One-tailed tests have one rejection region and, hence, only one critical value. The total α probability goes into that one side. The probability plots below display these values for right- and left-sided z-tests. These tests can detect effects in only one direction.

Related post : Understanding One-Tailed and Two-Tailed Hypothesis Tests and Effects in Statistics

Using a Critical Value to Construct Confidence Intervals

Confidence intervals use the same critical values (CVs) as the corresponding hypothesis test. The confidence level equals 1 – the significance level. Consequently, the CVs for a significance level of 0.05 produce a confidence level of 1 – 0.05 = 0.95 or 95%.

For example, to calculate the 95% confidence interval for our two-tailed z-test with a significance level of 0.05, use the CVs of -1.96 and 1.96 that we found above.

To calculate the upper and lower limits of the interval, take the positive critical value and multiply it by the standard error of the mean. Then take the sample mean and add and subtract that product from it.

• Lower Limit = Sample Mean – (CV * Standard Error of the Mean)
• Upper Limit = Sample Mean + (CV * Standard Error of the Mean)

To learn more about confidence intervals and how to construct them, read my posts about Confidence Intervals and How Confidence Intervals Work .

Related post : Standard Error of the Mean

How to Find a Critical Value

Unfortunately, the formulas for finding critical values are very complex. Typically, you don’t calculate them by hand. For the examples in this article, I’ve used statistical software to find them. However, you can also use statistical tables.

To learn how to use these critical value tables, read my articles that contain the tables and information about using them. The process for finding them is similar for the various tests. Using these tables requires knowing the correct test statistic, the significance level, the number of tails, and, in most cases, the degrees of freedom.

The following articles provide the statistical tables, explain how to use them, and visually illustrate the results.

• T distribution table
• Chi-square table

Related post : Degrees of Freedom

Critical Value Calculator

Another method for finding CVs is to use a critical value calculator, such as the one below. These calculators are handy for finding the answer, but they don’t provide the context for the results.

This calculator finds critical values for the sampling distributions of common test statistics.

For example, choose the following in the calculator:

• Z (standard normal)
• Significance level = 0.05

The calculator will display the same ±1.96 values we found earlier in this article.

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Reader Interactions

January 16, 2024 at 5:26 pm

Hello, I am currently taking statistics and am reviewing confidence intervals. I would like to know what is the equation for calculating a two-tailed test for upper and lower limits? I would like to know is there a way to calculate one and two-tailed tests without using a confidence interval calculator and can you explain further?

January 16, 2024 at 6:43 pm

If you’re talking about calculating the critical values values for a test statistic for two-tailed test, the calculations are fairly complex. Consequently, you’ll either use statistical software, an online calculator, or a statistical table to find those limits.

Comments and Questions Cancel reply

Critical Value Approach in Hypothesis Testing

After calculating the test statistic using the sample data, you compare it to the critical value(s) corresponding to the chosen significance level ( α ).

Two-sided test

Left-tailed test, right-tailed test, using critical values to construct confidence intervals.

Compute the lower bound and upper bound:

Finding the Critical Value

As you can see, the specific formula to find critical values depends on the distribution and the parameters associated with the problem at hand.

Take your skills to the next level ⚡️

P-Value vs. Critical Value: A Friendly Guide for Beginners

In the world of statistics, you may have come across the terms p-value and critical value . These concepts are essential in hypothesis testing, a process that helps you make informed decisions based on data. As you embark on your journey to understand the significance and applications of these values, don’t worry; you’re not alone. Many professionals and students alike grapple with these concepts, but once you get the hang of what they mean, they become powerful tools at your fingertips.

The main difference between p-value and critical value is that the p-value quantifies the strength of evidence against a null hypothesis, while the critical value sets a threshold for assessing the significance of a test statistic. Simply put, if your p-value is below the critical value, you reject the null hypothesis.

As you read on, you can expect to dive deeper into the definitions, applications, and interpretations of these often misunderstood statistical concepts. The remainder of the article will guide you through how p-values and critical values work in real-world scenarios, tips on interpreting their results, and potential pitfalls to avoid. By the end, you’ll have a clear understanding of their role in hypothesis testing, helping you become a more effective researcher or analyst.

Important Sidenote: We interviewed numerous data science professionals (data scientists, hiring managers, recruiters – you name it) and identified 6 proven steps to follow for becoming a data scientist. Read my article: ‘6 Proven Steps To Becoming a Data Scientist [Complete Guide] for in-depth findings and recommendations! – This is perhaps the most comprehensive article on the subject you will find on the internet!

Table of Contents

Understanding P-Value and Critical Value

When you dive into the world of statistics, it’s essential to grasp the concepts of P-value and critical value . These two values play a crucial role in hypothesis testing, helping you make informed decisions based on data. In this section, we will focus on the concept of hypothesis testing and how P-value and critical value relate to it.

Concept of Hypothesis Testing

Hypothesis testing is a statistical technique used to analyze data and draw conclusions. You start by creating a null hypothesis (H0) and an alternative hypothesis (H1). The null hypothesis represents the idea that there is no significant effect or relationship between the variables being tested, while the alternative hypothesis claims that there is a significant effect or relationship.

To conduct a hypothesis test, follow these steps:

• Formulate your null and alternative hypotheses.
• Choose an appropriate statistical test and significance level (α).
• Collect and analyze your data.
• Calculate the test statistic and P-value.
• Compare the P-value to the critical value.

Now, let’s discuss how P-value and critical value come into play during hypothesis testing.

The P-value is the probability of observing a test statistic as extreme (or more extreme) than the one calculated if the null hypothesis were true. In simpler terms, it’s the likelihood of getting your observed results by chance alone. The lower the P-value, the more evidence you have against the null hypothesis.

Here’s what you need to know about P-values:

• A low P-value (typically ≤ 0.05) indicates that the null hypothesis is unlikely to be true.
• A high P-value (typically > 0.05) suggests that the observed results align with the null hypothesis.

Critical Value

The critical value is a threshold that defines whether the test statistic is extreme enough to reject the null hypothesis. It depends on the chosen significance level (α) and the specific statistical test being used. If the test statistic exceeds the critical value, you reject the null hypothesis in favor of the alternative.

To summarize:

• If the P-value ≤ critical value, reject the null hypothesis.
• If the P-value > critical value, fail to reject the null hypothesis (do not conclude that the alternative is true).

In conclusion, understanding P-value and critical value is crucial for hypothesis testing. They help you determine the significance of your findings and make data-driven decisions. By grasping these concepts, you’ll be well-equipped to analyze data and draw meaningful conclusions in a variety of contexts.

P-Value Essentials

Calculating and interpreting p-values is essential to understanding statistical significance in research. In this section, we’ll cover the basics of p-values and how they relate to critical values.

Calculating P-Values

A p-value represents the probability of obtaining a result at least as extreme as the observed data, assuming the null hypothesis is correct. To calculate a p-value, follow these steps:

• Define your null and alternative hypotheses.
• Determine the test statistic and its distribution.
• Calculate the observed test statistic based on your sample data.
• Find the probability of obtaining a test statistic at least as extreme as the observed value.

Let’s dive deeper into these steps:

• Step 1: Formulate the null hypothesis (H₀) and alternative hypothesis (H₁). The null hypothesis typically states that there is no effect or relationship between variables, while the alternative hypothesis suggests otherwise.
• Step 2: Determine your test statistic and its distribution. The choice of test statistic depends on your data and hypotheses. Some common test statistics include the t -test, z -test, or chi-square test.
• Step 3: Using your sample data, compute the test statistic. This value quantifies the difference between your sample data and the null hypothesis.
• Step 4: Find the probability of obtaining a test statistic at least as extreme as the observed value, under the assumption that the null hypothesis is true. This probability is the p-value .

Interpreting P-Values

Once you’ve calculated the p-value, it’s time to interpret your results. The interpretation depends on the pre-specified significance level (α) you’ve chosen. Here’s a simplified guideline:

• If p-value ≤ α , you can reject the null hypothesis.
• If p-value > α , you cannot reject the null hypothesis.

Keep in mind that:

• A lower p-value indicates stronger evidence against the null hypothesis.
• A higher p-value implies weaker evidence against the null hypothesis.

Remember that statistical significance (p-value ≤ α) does not guarantee practical or scientific significance. It’s essential not to take the p-value as the sole metric for decision-making, but rather as a tool to help gauge your research outcomes.

In summary, p-values are crucial in understanding and interpreting statistical research results. By calculating and appropriately interpreting p-values, you can deepen your knowledge of your data and make informed decisions based on statistical evidence.

Critical Value Essentials

In this section, we’ll discuss two important aspects of critical values: Significance Level and Rejection Region . Knowing these concepts helps you better understand hypothesis testing and make informed decisions about the statistical significance of your results.

Significance Level

The significance level , often denoted as α or alpha, is an essential part of hypothesis testing. You can think of it as the threshold for deciding whether your results are statistically significant or not. In general, a common significance level is 0.05 or 5% , which means that there is a 5% chance of rejecting a true null hypothesis.

To help you understand better, here are a few key points:

• The lower the significance level, the more stringent the test.
• Higher α-levels may increase the risk of Type I errors (incorrectly rejecting the null hypothesis).
• Lower α-levels may increase the risk of Type II errors (failing to reject a false null hypothesis).

Rejection Region

The rejection region is the range of values that, if your test statistic falls within, leads to the rejection of the null hypothesis. This area depends on the critical value and the significance level. The critical value is a specific point that separates the rejection region from the rest of the distribution. Test statistics that fall in the rejection region provide evidence that the null hypothesis might not be true and should be rejected.

Here are essential points to consider when using the rejection region:

• Z-score : The z-score is a measure of how many standard deviations away from the mean a given value is. If your test statistic lies in the rejection region, it means that the z-score is significant.
• Rejection regions are tailored for both one-tailed and two-tailed tests.
• In a one-tailed test, the rejection region is either on the left or right side of the distribution.
• In a two-tailed test, there are two rejection regions, one on each side of the distribution.

By understanding and considering the significance level and rejection region, you can more effectively interpret your statistical results and avoid making false assumptions or claims. Remember that critical values are crucial in determining whether to reject or accept the null hypothesis.

Statistical Tests and Decision Making

When you’re comparing the means of two samples, a t-test is often used. This test helps you determine whether there is a significant difference between the means. Here’s how you can conduct a t-test:

• Calculate the t-statistic for your samples
• Determine the degrees of freedom
• Compare the t-statistic to a critical value from a t-distribution table

If the t-statistic is greater than the critical value, you can reject the null hypothesis and conclude that there is a significant difference between the sample means. Some key points about t-test:

• Test statistic : In a t-test, the t-statistic is the key value that you calculate
• Sample : For a t-test, you’ll need two independent samples to compare

The Analysis of Variance (ANOVA) is another statistical test, often used when you want to compare the means of three or more treatment groups. With this method, you analyze the differences between group means and make decisions on whether the total variation in the dataset can be accounted for by the variance within the groups or the variance between the groups. Here are the main steps in conducting an ANOVA test:

• Calculate the F statistic
• Determine the degrees of freedom for between-groups and within-groups
• Compare the F statistic to a critical value from an F-distribution table

When the F statistic is larger than the critical value, you can reject the null hypothesis and conclude that there is a significant difference among the treatment groups. Keep these points in mind for ANOVA tests:

• Treatment Groups : ANOVA tests require three or more groups to compare
• Observations : You need multiple observations within each treatment group

Confidence Intervals

Confidence intervals (CIs) are a way to estimate values within a certain range, with a specified level of confidence. They help to indicate the reliability of an estimated parameter, like the mean or difference between sample means. Here’s what you need to know about calculating confidence intervals:

• Determine the point estimate (e.g., sample mean or difference in means)
• Calculate the standard error
• Multiply the standard error by the appropriate critical value

The result gives you a range within which the true population parameter is likely to fall, with a certain level of confidence (e.g., 95%). Remember these insights when working with confidence intervals:

• Confidence Level : The confidence level is the probability that the true population parameter falls within the calculated interval
• Critical Value : Based on the specified confidence level, you’ll determine a critical value from a table (e.g., t-distribution)

Remember, using appropriate statistical tests, test statistics, and critical values will help you make informed decisions in your data analysis.

Comparing P-Values and Critical Values

Differences and Similarities

When analyzing data, you may come across two important concepts – p-values and critical values . While they both help determine the significance of a data set, they have some differences and similarities.

• P-values are probabilities, ranging from 0 to 1, indicating how likely it is a particular result could be observed if the null hypothesis is true. Lower p-values suggest the null hypothesis should be rejected, meaning the observed data is not due to chance alone.
• On the other hand, critical values are preset thresholds that decide whether the null hypothesis should be rejected or not. Results that surpass the critical value support adopting the alternative hypothesis.

The main similarity between p-values and critical values is their role in hypothesis testing. Both are used to determine if observed data provides enough evidence to reject the null hypothesis in favor of the alternative hypothesis.

Applications in Geospatial Data Analysis

In the field of geospatial data analysis, p-values and critical values play essential roles in making data-driven decisions. Researchers like Hartmann, Krois, and Waske from the Department of Earth Sciences at Freie Universitaet Berlin often use these concepts in their e-Learning project SOGA.

To better understand the applications, let’s look at three main aspects:

• Spatial autocorrelation : With geospatial data, points might be related not only by their values but also by their locations. P-values can help assess spatial autocorrelation and recognize underlying spatial patterns.
• Geostatistical analysis : Techniques like kriging or semivariogram estimation depend on critical values and p-values to decide the suitability of a model. By finding the best fit model, geospatial data can be better represented, ensuring accurate and precise predictions.
• Comparing geospatial data groups : When comparing two subsets of data (e.g., mineral concentrations, soil types), p-values can be used in permutation tests or t-tests to verify if the observed differences are significant or due to chance.

In summary, when working with geospatial data analysis, p-values and critical values are crucial tools that enable you to make informed decisions about your data and its implications. By understanding the differences and similarities between the two concepts, you can apply them effectively in your geospatial data analysis journey.

Standard Distributions and Scores

In this section, we will discuss the Standard Normal Distribution and its associated scores, namely Z-Score and T-Statistic . These concepts are crucial in understanding the differences between p-values and critical values.

Standard Normal Distribution

The Standard Normal Distribution is a probability distribution that has a mean of 0 and a standard deviation of 1. This distribution is crucial for hypothesis testing, as it helps you make inferences about your data based on standard deviations from the mean. Some characteristics of this distribution include:

• 68% of the data falls within ±1 standard deviation from the mean
• 95% of the data falls within ±2 standard deviations from the mean
• 99.7% of the data falls within ±3 standard deviations from the mean

The Z-Score is a measure of how many standard deviations away a data point is from the mean of the distribution. It is used to compare data points across different distributions with different means and standard deviations. To calculate the Z-Score, use the formula:

Key features of the Z-Score include:

• Positive Z-Scores indicate values above the mean
• Negative Z-Scores indicate values below the mean
• A Z-Score of 0 is equal to the mean

T-Statistic

The T-Statistic , also known as the Student’s t-distribution , is another way to assess how far away a data point is from the mean. It comes in handy when:

• You have a small sample size (generally less than 30)
• Population variance is not known
• Population is assumed to be normally distributed

The T-Statistic shares similarities with the Z-Score but adjusts for sample size, making it more appropriate for smaller samples. The formula for calculating the T-Statistic is:

In conclusion, understanding the Standard Normal Distribution , Z-Score , and T-Statistic will help you better differentiate between p-values and critical values, ultimately aiding in accurate statistical analysis and hypothesis testing.

Author’s Recommendations: Top Data Science Resources To Consider

Before concluding this article, I wanted to share few top data science resources that I have personally vetted for you. I am confident that you can greatly benefit in your data science journey by considering one or more of these resources.

• DataCamp: If you are a beginner focused towards building the foundational skills in data science , there is no better platform than DataCamp. Under one membership umbrella, DataCamp gives you access to 335+ data science courses. There is absolutely no other platform that comes anywhere close to this. Hence, if building foundational data science skills is your goal: Click Here to Sign Up For DataCamp Today!
• IBM Data Science Professional Certificate: If you are looking for a data science credential that has strong industry recognition but does not involve too heavy of an effort: Click Here To Enroll Into The IBM Data Science Professional Certificate Program Today! (To learn more: Check out my full review of this certificate program here )
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• Roadmap To Becoming a Data Scientist: If you have decided to become a data science professional but not fully sure how to get started : read my article – 6 Proven Ways To Becoming a Data Scientist . In this article, I share my findings from interviewing 100+ data science professionals at top companies (including – Google, Meta, Amazon, etc.) and give you a full roadmap to becoming a data scientist.

Frequently Asked Questions

What is the relationship between p-value and critical value.

The p-value represents the probability of observing the test statistic under the null hypothesis, while the critical value is a predetermined threshold for declaring significance. If the p-value is less than the critical value, you reject the null hypothesis.

How do you interpret p-value in comparison to critical value?

When the p-value is smaller than the critical value , there is strong evidence against the null hypothesis, which means you reject it. In contrast, if the p-value is larger, you fail to reject the null hypothesis and cannot conclude a significant effect.

What does it mean when the p-value is greater than the critical value?

If the p-value is greater than the critical value , it indicates that the observed data are consistent with the null hypothesis, and you do not have enough evidence to reject it. In other words, the finding is not statistically significant.

How are critical values used to determine significance?

Critical values are used as a threshold to determine if a test statistic is considered significant. When the test statistic is more extreme than the critical value, you reject the null hypothesis, indicating that the observed effect is unlikely due to chance alone.

Why is it important to know both p-value and critical value in hypothesis testing?

Knowing both p-value and critical value helps you to:

• Understand the strength of evidence against the null hypothesis
• Decide whether to reject or fail to reject the null hypothesis
• Assess the statistical significance of your findings
• Avoid misinterpretations and false conclusions

How do you calculate critical values and compare them to p-values?

To calculate critical values, you:

• Choose a significance level (α)
• Determine the appropriate test statistic distribution
• Find the value that corresponds to α in the distribution

Then, you compare the calculated critical value with the p-value to determine if the result is statistically significant or not. If the p-value is less than the critical value, you reject the null hypothesis.

BEFORE YOU GO: Don’t forget to check out my latest article – 6 Proven Steps To Becoming a Data Scientist [Complete Guide] . We interviewed numerous data science professionals (data scientists, hiring managers, recruiters – you name it) and created this comprehensive guide to help you land that perfect data science job.

Affiliate Disclosure: We participate in several affiliate programs and may be compensated if you make a purchase using our referral link, at no additional cost to you. You can, however, trust the integrity of our recommendation. Affiliate programs exist even for products that we are not recommending. We only choose to recommend you the products that we actually believe in.

Daisy is the founder of DataScienceNerd.com. Passionate for the field of Data Science, she shares her learnings and experiences in this domain, with the hope to help other Data Science enthusiasts in their path down this incredible discipline.

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Introduction to Statistics

Chapter 7 introduction to hypothesis testing, 7.1 chapter overview.

In this chapter, we will continue our discussion on statistical inference with a discussion on hypothesis testing. In hypothesis testing, we take a more active approach to our data by asking questions about population parameters and developing a framework to answer those questions. We will root this discussion in confidence intervals before learning about several other approaches to hypothesis testing.

Chapter Learning Outcomes/Objectives

• confidence intervals.
• the critical value approach.
• the p-value approach.

R Objectives

• Generate hypothesis tests for a mean.
• Interpret R output for tests of a mean.

This chapter’s outcomes correspond to course outcomes (6) apply statistical inference techniques of parameter estimation such as point estimation and confidence interval estimation and (7) apply techniques of testing various statistical hypotheses concerning population parameters.

7.2 Logic of Hypothesis Testing

This section is framed in terms of questions about a population mean \(\mu\) , but the same logic applies to \(p\) (and other population parameters).

One of our goals with statistical inference is to make decisions or judgements about the value of a parameter. A confidence interval is a good starting point, but we might also want to ask questions like

• Do cans of soda actually contain 12 oz?
• Is Medicine A better than Medicine B?

A hypothesis is a statement that something is true. A hypothesis test involves two (competing) hypotheses:

• The null hypothesis , denoted \(H_0\) , is the hypothesis to be tested. This is the “default” assumption.
• The alternative hypothesis , denoted \(H_A\) is the alternative to the null.

Note that the subscript 0 is “nought” (pronounced “not”). A hypothesis test helps us decide whether the null hypothesis should be rejected in favor of the alternative.

Example : Cans of soda are labeled with “12 FL OZ”. Is this accurate? The default, or uninteresting, assumption is that cans of soda contain 12 oz. \(H_0\) : the mean volume of soda in a can is 12 oz. \(H_A\) : the mean volume of soda in a can is NOT 12 oz.

We can write these hypotheses in words (as above) or in statistical notation. The null specifies a single value of \(\mu\)

• \(H_0\) : \(\mu=\mu_0\)

We call \(\mu_0\) the null value . When we run a hypothesis test, \(\mu_0\) will be replaced by some number. For the soda can example, the null value is 12. We would write \(H_0: \mu = 12\) .

The alternative specifies a range of possible values for \(\mu\) :

• \(H_A\) : \(\mu\ne\mu_0\) . “The true mean is different from the null value.”

Take a random sample from the population. If the data area consistent with the null hypothesis, do not reject the null hypothesis. If the data are inconsistent with the null hypothesis and supportive of the alternative hypothesis, reject the null in favor of the alternative.

Example : One way to think about the logic of hypothesis testing is by comparing it to the U.S. court system. In a jury trial, jurors are told to assume the defendant is “innocent until proven guilty”. Innocence is the default assumption, so \(H_0\) : the defendant is innocent. \(H_A\) : the defendant is guilty. Like in hypothesis testing, it is not the jury’s job to decide if the defendant is innocent. That should be their default assumption. They are only there to decide if the defendant is guilty or if there is not enough evidence to override that default assumption. The burden of proof lies on the alternative hypothesis.

Notice the careful language in the logic of hypothesis testing: we either reject, or fail to reject, the null hypothesis. We never “accept” a null hypothesis.

7.2.1 Decision Errors

• A Type I Error is rejecting the null when it is true. (Null is true, but we conclude null is false.)
• A Type II Error is not rejecting the null when it is false. (Null is false, but we do not conclude it is false.)

Example : In our jury trial, \(H_0\) : the defendant is innocent. \(H_A\) : the defendant is guilty. A Type I error is concluding guilt when the defendant is innocent. A Type II error is failing to convict when the person is guilty.

How likely are we to make errors? Well, \(P(\) Type I Error \()=\alpha\) , the significance level . (Yes, this is the same \(\alpha\) we saw in confidence intervals!) For Type II error, \(P(\) Type II Error \()=\beta\) . This is related to the sample size calculation from the previous chapter, but is otherwise something we don’t have time to cover.

We would like both \(\alpha\) and \(\beta\) to be small but, like many other things in statistics, there’s a trade off! For a fixed sample size,

• If we decrease \(\alpha\) , then \(\beta\) will increase.
• If we increase \(\alpha\) , then \(\beta\) will decrease.

In practice, we set \(\alpha\) (as we did in confidence intervals). We can improve \(\beta\) by increasing sample size. Since resources are finite (we can’t get enormous sample sizes all the time), we will need to consider the consequences of each type of error.

Example We could think about assessing consequences through the jury trial example. Consider two possible charges: Defendant is accused of stealing a loaf of bread. If found guilty, they may face some jail time and will have a criminal record. Defendant is accused of murder. If found guilty, they will have a felony and may spend decades in prison. Since these are moral questions, I will let you consider the consequences of each type of error. However, keep in mind that we do make scientific decisions that have lasting impacts on people’s lives.

• At the \(\alpha\) level of significance, the data provide sufficient evidence to support the alternative hypothesis.
• At the \(\alpha\) level of significance, the data do not provide sufficient evidence to support the alternative hypothesis.

Notice that these conclusions are framed in terms of the alternative hypothesis, which is either supported or not supported. We will never conclude the null hypothesis. Finally, when we write these types of conclusions, we will write them in the context of the problem.

7.3 Confidence Interval Approach to Hypothesis Testing

We can use a confidence interval to help us weigh the evidence against the null hypothesis. A confidence interval gives us a range of plausible values for \(\mu\) . If the null value is in the interval, then \(\mu_0\) is a plausible value for \(\mu\) . If the null value is not in the interval, then \(\mu_0\) is not a plausible value for \(\mu\) .

• State null and alternative hypotheses.
• Decide on significance level \(\alpha\) . Check assumptions (decide which confidence interval setting to use).
• Find the critical value.
• Compute confidence interval.
• If the null value is not in the confidence interval, reject the null hypothesis. Otherwise, do not reject.
• Interpret results in the context of the problem.

Example : Is the average mercury level in dolphin muslces different from \(2.5\mu g/g\) ? Test at the 0.05 level of significance. A random sample of \(19\) dolphins resulted in a mean of \(4.4 \mu g/g\) and a standard deviation of \(2.3 \mu g/g\) . \(H_0: \mu = 2.5\) and \(H_A: \mu \ne 2.5\) . Significance level is \(\alpha=0.05\) . The value of \(\sigma\) is unknown and \(n = 19 < 30\) , so we are in setting 3. For setting 3, the critical value is \(t_{df, \alpha/2}\) . Here, \(df=n-1=18\) and \(\alpha/2 = 0.025\) :

The confidence interval is \[\begin{align} \bar{x} &\pm t_{df, \alpha/2}\frac{s}{\sqrt{n}} \\ 4.4 &\pm 2.101 \frac{2.3}{\sqrt{19}} \\ 4.4 &\pm 1.109 \end{align}\] or \((3.29, 5.51)\) . Since the null value, \(2.5\) , is not in the interval, it is not a plausible value for \(\mu\) (at the 95% level of confidence). Therefore we reject the null hypothesis. At the 0.05 level of significance, the data provide sufficient evidence to conclude that the true mean mercury level in dolphin muscles is greater than \(2.5\mu g/g\) . Note: The alternative hypothesis is “not equal to”, but we conclude “greater than” because all of the plausible values in the confidence interval are greater than the null value.

7.4 Critical Value Approach to Hypothesis Testing

We learned about critical values when we discussed confidence intervals. Now, we want to use these values directly in a hypothesis test. We will compare these values to a value based on the data, called a test statistic .

Idea: the null is our “default assumption”. If the null is true, how likely are we to observe a sample that looks like the one we have? If our sample is very inconsistent with the null hypothesis, we want to reject the null hypothesis.

7.4.1 Test statistics

Test statistics are similar to z- and t-scores: \[\text{test statistic} = \frac{\text{point estimate}-\text{null value}}{\text{standard error}}.\] In fact, they serve a similar function in converting a variable \(\bar{X}\) into a distribution we can work with easily.

• Large Sample Setting : \(\mu\) is target parameter, \(n \ge 30\)

\[z = \frac{\bar{x}-\mu_0}{s/\sqrt{n}}\]

• Small Sample Setting : \(\mu\) is target parameter, \(n < 30\)

\[t = \frac{\bar{x}-\mu_0}{s/\sqrt{n}}\]

The set of values for the test statistic that cause us to reject \(H_0\) is the rejection region . The remaining values are the nonrejection region . The value that separates these is the critical value!

• State the null and alternative hypotheses.
• Determine the significance level \(\alpha\) . Check assumptions (decide which setting to use).
• Compute the value of the test statistic.
• Determine the critical values.
• If the test statistic is in the rejection region, reject the null hypothesis. Otherwise, do not reject.
• Interpret results.

Example : Is the average mercury level in dolphin muslces different from \(2.5\mu g/g\) ? Test at the 0.05 level of significance. A random sample of \(19\) dolphins resulted in a mean of \(4.4 \mu g/g\) and a standard deviation of \(2.3 \mu g/g\) . \(H_0: \mu = 2.5\) and \(H_A: \mu \ne 2.5\) . Significance level is \(\alpha=0.05\) . The value of \(\sigma\) is unknown and \(n = 19 < 30\) , so we are in setting 3. The test statistic is \[\begin{align} t &= \frac{\bar{x}-\mu_0}{s/\sqrt{n}} \\ &= \frac{4.4-2.5}{2.3/\sqrt{19}} \\ &= 3.601 \end{align}\] The critical value is \(t_{df, \alpha/2}\) . Here, \(df=n-1=18\) and \(\alpha/2 = 0.025\) :

The test statistic is in the rejection region, so we will reject the null hypothesis:

At the 0.05 level of significance, the data provide sufficient evidence to conclude that the true mean mercury level in dolphin muscles is greater than \(2.5\mu g/g\) .

Notice that this is the same conclusion we came to when we used the confidence interval approach. These approaches are exactly equivalent!

7.5 P-Value Approach to Hypothesis Testing

If the null hypothesis is true, what is the probability of getting a random sample that is as inconsistent with the null hypothesis as the random sample we got? This probability is called the p-value .

Example : Is the average mercury level in dolphin muscles different from \(2.5\mu g/g\) ? Test at the 0.05 level of significance. A random sample of \(19\) dolphins resulted in a mean of \(4.4 \mu g/g\) and a standard deviation of \(2.3 \mu g/g\) . Probability of a sample as inconsistent as our sample is \(P(t_{df} \text{ is as extreme as the test statistic})\) . Consider \[P(t_{18} > 3.6) = 0.001\] but we want to think about the probability of being “as extreme” in either direction (either tail), so \[\text{p-value} = 2P(t_{18}>3.6) = 0.002\]

• If \(\text{p-value} < \alpha\) , reject the null hypothesis. Otherwise, do not reject.

7.5.1 P-Values

Large Sample Setting : \(\mu\) is target parameter, \(n \ge 30\) , \[2P(Z > |z|)\] where \(z\) is the test statistic.

Small Sample Setting : \(\mu\) is target parameter, \(n < 30\) , \[2P(t_{df} > |t|)\] where \(t\) is the test statistic.

Note: \(|a|\) is the “absolute value” of \(a\) . The absolute value takes a number and throws away the sign, so \(|2|=2\) and \(|-3|=3\) .

• Determine the p-value.

We often use p-values instead of the critical value approach because they are meaningful on their own (they have a direct interpretation).

Example : For the dolphins, \(H_0: \mu = 2.5\) and \(H_A: \mu \ne 2.5\) . Significance level is \(\alpha=0.05\) . The value of \(\sigma\) is unknown and \(n = 19 < 30\) , so we are in setting 3. The test statistic is \[\begin{align} t &= \frac{\bar{x}-\mu_0}{s/\sqrt{n}} \\ &= \frac{4.4-2.5}{2.3/\sqrt{19}} \\ &= 3.601 \end{align}\] The p-value is \[2P(t_{df} > |t|) - 2P(t_{18} > 3.601) = 0.002\] Since \(\text{p-value}=0.002 < \alpha=0.05\) , reject the null hypothesis. At the 0.05 level of significance, the data provide sufficient evidence to conclude that the true mean mercury level in dolphin muscles is greater than \(2.5\mu g/g\) .

As before, this is the same conclusion we came to when we used the confidence interval and critical value approaches. All of these approaches are exactly equivalent.

R: Hypothesis Tests for a Mean

To conduct hypothesis tests for a mean in R, we will again use the t.test command. The arguments we will use for hypothesis testing are

• x : the variable that contains the data we want to use to construct a confidence interval.
• mu : the null value, \(\mu_0\) .
• conf.level : the desired confidence level ( \(1-\alpha\) ).

We will again to use the Loblolly pine tree data.

Let’s test if the average height of Loblolly pines differs from \(40\) feet. We will test at a 0.01 level of significance ( \(\alpha = 0.01\) ). So \(H_0: \mu = 40\) and \(H_A: \mu \ne 40\) and the R command will look like

Last time we used this command, we noted that R printed more information than we knew how to handle. That information was about hypothesis tests! The output from this test shows the following (top to bottom):

• the data used in the hypothesis test.
• the value of the test statistic ( \(t = -3.3851\) ), the degrees of freedom ( \(83\) ), and the p-value ( \(0.001\) ).
• the alternative hypothesis.
• the confidence interval.
• the sample mean.

Based on this output, we have everything we need to conduct a hypothesis test using (A) the confidence interval approach, (B) the critical value approach, or (C) the p-value approach! In practice, we might include results from multiple approaches: At the 0.01 level of significance, there is sufficient evidence to reject the null hypothesis and conclude that the true mean height of Loblolly pines is less than 40 feet ( \(t = -3.385\) , p-value \(=0.001\) ).

What is a critical value?

A critical value is a point on the distribution of the test statistic under the null hypothesis that defines a set of values that call for rejecting the null hypothesis. This set is called critical or rejection region. Usually, one-sided tests have one critical value and two-sided test have two critical values. The critical values are determined so that the probability that the test statistic has a value in the rejection region of the test when the null hypothesis is true equals the significance level (denoted as α or alpha).

Critical values on the standard normal distribution for α = 0.05

Figure A shows that results of a one-tailed Z-test are significant if the value of the test statistic is equal to or greater than 1.64, the critical value in this case. The shaded area represents the probability of a type I error (α = 5% in this example) of the area under the curve. Figure B shows that results of a two-tailed Z-test are significant if the absolute value of the test statistic is equal to or greater than 1.96, the critical value in this case. The two shaded areas sum to 5% (α) of the area under the curve.

Examples of calculating critical values

In hypothesis testing, there are two ways to determine whether there is enough evidence from the sample to reject H 0 or to fail to reject H 0 . The most common way is to compare the p-value with a pre-specified value of α, where α is the probability of rejecting H 0 when H 0 is true. However, an equivalent approach is to compare the calculated value of the test statistic based on your data with the critical value. The following are examples of how to calculate the critical value for a 1-sample t-test and a One-Way ANOVA.

Calculating a critical value for a 1-sample t-test

• Select Calc > Probability Distributions > t .
• Select Inverse cumulative probability .
• In Degrees of freedom , enter 9 (the number of observations minus one).
• In Input constant , enter 0.95 (one minus one-half alpha).

This gives you an inverse cumulative probability, which equals the critical value, of 1.83311. If the absolute value of the t-statistic value is greater than this critical value, then you can reject the null hypothesis, H 0 , at the 0.10 level of significance.

Calculating a critical value for an analysis of variance (ANOVA)

• Choose Calc > Probability Distributions > F .
• In Numerator degrees of freedom , enter 2 (the number of factor levels minus one).
• In Denominator degrees of freedom , enter 9 (the degrees of freedom for error).
• In Input constant , enter 0.95 (one minus alpha).

This gives you an inverse cumulative probability (critical value) of 4.25649. If the F-statistic is greater than this critical value, then you can reject the null hypothesis, H 0 , at the 0.05 level of significance.

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Statistics Tutorial

Descriptive statistics, inferential statistics, stat reference, statistics - hypothesis testing.

Hypothesis testing is a formal way of checking if a hypothesis about a population is true or not.

Hypothesis Testing

A hypothesis is a claim about a population parameter .

A hypothesis test is a formal procedure to check if a hypothesis is true or not.

Examples of claims that can be checked:

The average height of people in Denmark is more than 170 cm.

The share of left handed people in Australia is not 10%.

The average income of dentists is less the average income of lawyers.

The Null and Alternative Hypothesis

Hypothesis testing is based on making two different claims about a population parameter.

The null hypothesis (\(H_{0} \)) and the alternative hypothesis (\(H_{1}\)) are the claims.

The two claims needs to be mutually exclusive , meaning only one of them can be true.

The alternative hypothesis is typically what we are trying to prove.

For example, we want to check the following claim:

"The average height of people in Denmark is more than 170 cm."

In this case, the parameter is the average height of people in Denmark (\(\mu\)).

The null and alternative hypothesis would be:

Null hypothesis : The average height of people in Denmark is 170 cm.

Alternative hypothesis : The average height of people in Denmark is more than 170 cm.

The claims are often expressed with symbols like this:

\(H_{0}\): \(\mu = 170 \: cm \)

\(H_{1}\): \(\mu > 170 \: cm \)

If the data supports the alternative hypothesis, we reject the null hypothesis and accept the alternative hypothesis.

If the data does not support the alternative hypothesis, we keep the null hypothesis.

Note: The alternative hypothesis is also referred to as (\(H_{A} \)).

The Significance Level

The significance level (\(\alpha\)) is the uncertainty we accept when rejecting the null hypothesis in the hypothesis test.

The significance level is a percentage probability of accidentally making the wrong conclusion.

Typical significance levels are:

• \(\alpha = 0.1\) (10%)
• \(\alpha = 0.05\) (5%)
• \(\alpha = 0.01\) (1%)

A lower significance level means that the evidence in the data needs to be stronger to reject the null hypothesis.

There is no "correct" significance level - it only states the uncertainty of the conclusion.

Note: A 5% significance level means that when we reject a null hypothesis:

We expect to reject a true null hypothesis 5 out of 100 times.

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The Test Statistic

The test statistic is used to decide the outcome of the hypothesis test.

The test statistic is a standardized value calculated from the sample.

Standardization means converting a statistic to a well known probability distribution .

The type of probability distribution depends on the type of test.

Common examples are:

• Standard Normal Distribution (Z): used for Testing Population Proportions
• Student's T-Distribution (T): used for Testing Population Means

Note: You will learn how to calculate the test statistic for each type of test in the following chapters.

The Critical Value and P-Value Approach

There are two main approaches used for hypothesis tests:

• The critical value approach compares the test statistic with the critical value of the significance level.
• The p-value approach compares the p-value of the test statistic and with the significance level.

The Critical Value Approach

The critical value approach checks if the test statistic is in the rejection region .

The rejection region is an area of probability in the tails of the distribution.

The size of the rejection region is decided by the significance level (\(\alpha\)).

The value that separates the rejection region from the rest is called the critical value .

Here is a graphical illustration:

If the test statistic is inside this rejection region, the null hypothesis is rejected .

For example, if the test statistic is 2.3 and the critical value is 2 for a significance level (\(\alpha = 0.05\)):

We reject the null hypothesis (\(H_{0} \)) at 0.05 significance level (\(\alpha\))

The P-Value Approach

The p-value approach checks if the p-value of the test statistic is smaller than the significance level (\(\alpha\)).

The p-value of the test statistic is the area of probability in the tails of the distribution from the value of the test statistic.

If the p-value is smaller than the significance level, the null hypothesis is rejected .

The p-value directly tells us the lowest significance level where we can reject the null hypothesis.

For example, if the p-value is 0.03:

We reject the null hypothesis (\(H_{0} \)) at a 0.05 significance level (\(\alpha\))

We keep the null hypothesis (\(H_{0}\)) at a 0.01 significance level (\(\alpha\))

Note: The two approaches are only different in how they present the conclusion.

Steps for a Hypothesis Test

The following steps are used for a hypothesis test:

• Check the conditions
• Define the claims
• Decide the significance level
• Calculate the test statistic

One condition is that the sample is randomly selected from the population.

The other conditions depends on what type of parameter you are testing the hypothesis for.

Common parameters to test hypotheses are:

• Proportions (for qualitative data)
• Mean values (for numerical data)

You will learn the steps for both types in the following pages.

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The Critical Value and the p-Value Approach to Hypothesis Testing

Critical Value Calculator

How to use critical value calculator, what is a critical value, critical value definition, how to calculate critical values, z critical values, t critical values, chi-square critical values (χ²), f critical values, behind the scenes of the critical value calculator.

Welcome to the critical value calculator! Here you can quickly determine the critical value(s) for two-tailed tests, as well as for one-tailed tests. It works for most common distributions in statistical testing: the standard normal distribution N(0,1) (that is when you have a Z-score), t-Student, chi-square, and F-distribution .

What is a critical value? And what is the critical value formula? Scroll down – we provide you with the critical value definition and explain how to calculate critical values in order to use them to construct rejection regions (also known as critical regions).

The critical value calculator is your go-to tool for swiftly determining critical values in statistical tests, be it one-tailed or two-tailed. To effectively use the calculator, follow these steps:

In the first field, input the distribution of your test statistic under the null hypothesis: is it a standard normal N (0,1), t-Student, chi-squared, or Snedecor's F? If you are not sure, check the sections below devoted to those distributions, and try to localize the test you need to perform.

In the field What type of test? choose the alternative hypothesis : two-tailed, right-tailed, or left-tailed.

If needed, specify the degrees of freedom of the test statistic's distribution. If you need more clarification, check the description of the test you are performing. You can learn more about the meaning of this quantity in statistics from the degrees of freedom calculator .

Set the significance level, α \alpha α . By default, we pre-set it to the most common value, 0.05, but you can adjust it to your needs.

The critical value calculator will display your critical value(s) and the rejection region(s).

Click the advanced mode if you need to increase the precision with which the critical values are computed.

For example, let's envision a scenario where you are conducting a one-tailed hypothesis test using a t-Student distribution with 15 degrees of freedom. You have opted for a right-tailed test and set a significance level (α) of 0.05. The results indicate that the critical value is 1.7531, and the critical region is (1.7531, ∞). This implies that if your test statistic exceeds 1.7531, you will reject the null hypothesis at the 0.05 significance level.

👩‍🏫 Want to learn more about critical values? Keep reading!

In hypothesis testing, critical values are one of the two approaches which allow you to decide whether to retain or reject the null hypothesis. The other approach is to calculate the p-value (for example, using the p-value calculator ).

The critical value approach consists of checking if the value of the test statistic generated by your sample belongs to the so-called rejection region , or critical region , which is the region where the test statistic is highly improbable to lie . A critical value is a cut-off value (or two cut-off values in the case of a two-tailed test) that constitutes the boundary of the rejection region(s). In other words, critical values divide the scale of your test statistic into the rejection region and the non-rejection region.

Once you have found the rejection region, check if the value of the test statistic generated by your sample belongs to it :

• If so, it means that you can reject the null hypothesis and accept the alternative hypothesis; and
• If not, then there is not enough evidence to reject H 0 .

But how to calculate critical values? First of all, you need to set a significance level , α \alpha α , which quantifies the probability of rejecting the null hypothesis when it is actually correct. The choice of α is arbitrary; in practice, we most often use a value of 0.05 or 0.01. Critical values also depend on the alternative hypothesis you choose for your test , elucidated in the next section .

To determine critical values, you need to know the distribution of your test statistic under the assumption that the null hypothesis holds. Critical values are then points with the property that the probability of your test statistic assuming values at least as extreme at those critical values is equal to the significance level α . Wow, quite a definition, isn't it? Don't worry, we'll explain what it all means.

First, let us point out it is the alternative hypothesis that determines what "extreme" means. In particular, if the test is one-sided, then there will be just one critical value; if it is two-sided, then there will be two of them: one to the left and the other to the right of the median value of the distribution.

Critical values can be conveniently depicted as the points with the property that the area under the density curve of the test statistic from those points to the tails is equal to α \alpha α :

Left-tailed test: the area under the density curve from the critical value to the left is equal to α \alpha α ;

Right-tailed test: the area under the density curve from the critical value to the right is equal to α \alpha α ; and

Two-tailed test: the area under the density curve from the left critical value to the left is equal to α / 2 \alpha/2 α /2 , and the area under the curve from the right critical value to the right is equal to α / 2 \alpha/2 α /2 as well; thus, total area equals α \alpha α .

As you can see, finding the critical values for a two-tailed test with significance α \alpha α boils down to finding both one-tailed critical values with a significance level of α / 2 \alpha/2 α /2 .

The formulae for the critical values involve the quantile function , Q Q Q , which is the inverse of the cumulative distribution function ( c d f \mathrm{cdf} cdf ) for the test statistic distribution (calculated under the assumption that H 0 holds!): Q = c d f − 1 Q = \mathrm{cdf}^{-1} Q = cdf − 1 .

Once we have agreed upon the value of α \alpha α , the critical value formulae are the following:

• Left-tailed test :
• Right-tailed test :
• Two-tailed test :

In the case of a distribution symmetric about 0 , the critical values for the two-tailed test are symmetric as well:

Unfortunately, the probability distributions that are the most widespread in hypothesis testing have somewhat complicated c d f \mathrm{cdf} cdf formulae. To find critical values by hand, you would need to use specialized software or statistical tables. In these cases, the best option is, of course, our critical value calculator! 😁

Use the Z (standard normal) option if your test statistic follows (at least approximately) the standard normal distribution N(0,1) .

In the formulae below, u u u denotes the quantile function of the standard normal distribution N(0,1):

Left-tailed Z critical value: u ( α ) u(\alpha) u ( α )

Right-tailed Z critical value: u ( 1 − α ) u(1-\alpha) u ( 1 − α )

Two-tailed Z critical value: ± u ( 1 − α / 2 ) \pm u(1- \alpha/2) ± u ( 1 − α /2 )

Check out Z-test calculator to learn more about the most common Z-test used on the population mean. There are also Z-tests for the difference between two population means, in particular, one between two proportions.

Use the t-Student option if your test statistic follows the t-Student distribution . This distribution is similar to N(0,1) , but its tails are fatter – the exact shape depends on the number of degrees of freedom . If this number is large (>30), which generically happens for large samples, then the t-Student distribution is practically indistinguishable from N(0,1). Check our t-statistic calculator to compute the related test statistic.

In the formulae below, Q t , d Q_{\text{t}, d} Q t , d ​ is the quantile function of the t-Student distribution with d d d degrees of freedom:

Left-tailed t critical value: Q t , d ( α ) Q_{\text{t}, d}(\alpha) Q t , d ​ ( α )

Right-tailed t critical value: Q t , d ( 1 − α ) Q_{\text{t}, d}(1 - \alpha) Q t , d ​ ( 1 − α )

Two-tailed t critical values: ± Q t , d ( 1 − α / 2 ) \pm Q_{\text{t}, d}(1 - \alpha/2) ± Q t , d ​ ( 1 − α /2 )

Visit the t-test calculator to learn more about various t-tests: the one for a population mean with an unknown population standard deviation , those for the difference between the means of two populations (with either equal or unequal population standard deviations), as well as about the t-test for paired samples .

Use the χ² (chi-square) option when performing a test in which the test statistic follows the χ²-distribution .

You need to determine the number of degrees of freedom of the χ²-distribution of your test statistic – below, we list them for the most commonly used χ²-tests.

Here we give the formulae for chi square critical values; Q χ 2 , d Q_{\chi^2, d} Q χ 2 , d ​ is the quantile function of the χ²-distribution with d d d degrees of freedom:

Left-tailed χ² critical value: Q χ 2 , d ( α ) Q_{\chi^2, d}(\alpha) Q χ 2 , d ​ ( α )

Right-tailed χ² critical value: Q χ 2 , d ( 1 − α ) Q_{\chi^2, d}(1 - \alpha) Q χ 2 , d ​ ( 1 − α )

Two-tailed χ² critical values: Q χ 2 , d ( α / 2 ) Q_{\chi^2, d}(\alpha/2) Q χ 2 , d ​ ( α /2 ) and Q χ 2 , d ( 1 − α / 2 ) Q_{\chi^2, d}(1 - \alpha/2) Q χ 2 , d ​ ( 1 − α /2 )

Several different tests lead to a χ²-score:

Goodness-of-fit test : does the empirical distribution agree with the expected distribution?

This test is right-tailed . Its test statistic follows the χ²-distribution with k − 1 k - 1 k − 1 degrees of freedom, where k k k is the number of classes into which the sample is divided.

Independence test : is there a statistically significant relationship between two variables?

This test is also right-tailed , and its test statistic is computed from the contingency table. There are ( r − 1 ) ( c − 1 ) (r - 1)(c - 1) ( r − 1 ) ( c − 1 ) degrees of freedom, where r r r is the number of rows, and c c c is the number of columns in the contingency table.

Test for the variance of normally distributed data : does this variance have some pre-determined value?

This test can be one- or two-tailed! Its test statistic has the χ²-distribution with n − 1 n - 1 n − 1 degrees of freedom, where n n n is the sample size.

Finally, choose F (Fisher-Snedecor) if your test statistic follows the F-distribution . This distribution has a pair of degrees of freedom .

Let us see how those degrees of freedom arise. Assume that you have two independent random variables, X X X and Y Y Y , that follow χ²-distributions with d 1 d_1 d 1 ​ and d 2 d_2 d 2 ​ degrees of freedom, respectively. If you now consider the ratio ( X d 1 ) : ( Y d 2 ) (\frac{X}{d_1}):(\frac{Y}{d_2}) ( d 1 ​ X ​ ) : ( d 2 ​ Y ​ ) , it turns out it follows the F-distribution with ( d 1 , d 2 ) (d_1, d_2) ( d 1 ​ , d 2 ​ ) degrees of freedom. That's the reason why we call d 1 d_1 d 1 ​ and d 2 d_2 d 2 ​ the numerator and denominator degrees of freedom , respectively.

In the formulae below, Q F , d 1 , d 2 Q_{\text{F}, d_1, d_2} Q F , d 1 ​ , d 2 ​ ​ stands for the quantile function of the F-distribution with ( d 1 , d 2 ) (d_1, d_2) ( d 1 ​ , d 2 ​ ) degrees of freedom:

Left-tailed F critical value: Q F , d 1 , d 2 ( α ) Q_{\text{F}, d_1, d_2}(\alpha) Q F , d 1 ​ , d 2 ​ ​ ( α )

Right-tailed F critical value: Q F , d 1 , d 2 ( 1 − α ) Q_{\text{F}, d_1, d_2}(1 - \alpha) Q F , d 1 ​ , d 2 ​ ​ ( 1 − α )

Two-tailed F critical values: Q F , d 1 , d 2 ( α / 2 ) Q_{\text{F}, d_1, d_2}(\alpha/2) Q F , d 1 ​ , d 2 ​ ​ ( α /2 ) and Q F , d 1 , d 2 ( 1 − α / 2 ) Q_{\text{F}, d_1, d_2}(1 -\alpha/2) Q F , d 1 ​ , d 2 ​ ​ ( 1 − α /2 )

Here we list the most important tests that produce F-scores: each of them is right-tailed .

ANOVA : tests the equality of means in three or more groups that come from normally distributed populations with equal variances. There are ( k − 1 , n − k ) (k - 1, n - k) ( k − 1 , n − k ) degrees of freedom, where k k k is the number of groups, and n n n is the total sample size (across every group).

Overall significance in regression analysis . The test statistic has ( k − 1 , n − k ) (k - 1, n - k) ( k − 1 , n − k ) degrees of freedom, where n n n is the sample size, and k k k is the number of variables (including the intercept).

Compare two nested regression models . The test statistic follows the F-distribution with ( k 2 − k 1 , n − k 2 ) (k_2 - k_1, n - k_2) ( k 2 ​ − k 1 ​ , n − k 2 ​ ) degrees of freedom, where k 1 k_1 k 1 ​ and k 2 k_2 k 2 ​ are the number of variables in the smaller and bigger models, respectively, and n n n is the sample size.

The equality of variances in two normally distributed populations . There are ( n − 1 , m − 1 ) (n - 1, m - 1) ( n − 1 , m − 1 ) degrees of freedom, where n n n and m m m are the respective sample sizes.

I'm Anna, the mastermind behind the critical value calculator and a PhD in mathematics from Jagiellonian University .

The idea for creating the tool originated from my experiences in teaching and research. Recognizing the need for a tool that simplifies the critical value determination process across various statistical distributions, I built a user-friendly calculator accessible to both students and professionals. After publishing the tool, I soon found myself using the calculator in my research and as a teaching aid.

Trust in this calculator is paramount to me. Each tool undergoes a rigorous review process , with peer-reviewed insights from experts and meticulous proofreading by native speakers. This commitment to accuracy and reliability ensures that users can be confident in the content. Please check the Editorial Policies page for more details on our standards.

What is a Z critical value?

A Z critical value is the value that defines the critical region in hypothesis testing when the test statistic follows the standard normal distribution . If the value of the test statistic falls into the critical region, you should reject the null hypothesis and accept the alternative hypothesis.

How do I calculate Z critical value?

To find a Z critical value for a given confidence level α :

Check if you perform a one- or two-tailed test .

For a one-tailed test:

Left -tailed: critical value is the α -th quantile of the standard normal distribution N(0,1).

Right -tailed: critical value is the (1-α) -th quantile.

Two-tailed test: critical value equals ±(1-α/2) -th quantile of N(0,1).

No quantile tables ? Use CDF tables! (The quantile function is the inverse of the CDF.)

Verify your answer with an online critical value calculator.

Is a t critical value the same as Z critical value?

In theory, no . In practice, very often, yes . The t-Student distribution is similar to the standard normal distribution, but it is not the same . However, if the number of degrees of freedom (which is, roughly speaking, the size of your sample) is large enough (>30), then the two distributions are practically indistinguishable , and so the t critical value has practically the same value as the Z critical value.

What is the Z critical value for 95% confidence?

The Z critical value for a 95% confidence interval is:

• 1.96 for a two-tailed test;
• 1.64 for a right-tailed test; and
• -1.64 for a left-tailed test.

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9.3 - the p-value approach, example 9-4 section  .

Up until now, we have used the critical region approach in conducting our hypothesis tests. Now, let's take a look at an example in which we use what is called the P -value approach .

Among patients with lung cancer, usually, 90% or more die within three years. As a result of new forms of treatment, it is felt that this rate has been reduced. In a recent study of n = 150 lung cancer patients, y = 128 died within three years. Is there sufficient evidence at the \(\alpha = 0.05\) level, say, to conclude that the death rate due to lung cancer has been reduced?

The sample proportion is:

\(\hat{p}=\dfrac{128}{150}=0.853\)

The null and alternative hypotheses are:

\(H_0 \colon p = 0.90\) and \(H_A \colon p < 0.90\)

The test statistic is, therefore:

\(Z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}=\dfrac{0.853-0.90}{\sqrt{\dfrac{0.90(0.10)}{150}}}=-1.92\)

And, the rejection region is:

Since the test statistic Z = −1.92 < −1.645, we reject the null hypothesis. There is sufficient evidence at the \(\alpha = 0.05\) level to conclude that the rate has been reduced.

Example 9-4 (continued) Section  

What if we set the significance level \(\alpha\) = P (Type I Error) to 0.01? Is there still sufficient evidence to conclude that the death rate due to lung cancer has been reduced?

In this case, with \(\alpha = 0.01\), the rejection region is Z ≤ −2.33. That is, we reject if the test statistic falls in the rejection region defined by Z ≤ −2.33:

Because the test statistic Z = −1.92 > −2.33, we do not reject the null hypothesis. There is insufficient evidence at the \(\alpha = 0.01\) level to conclude that the rate has been reduced.

In the first part of this example, we rejected the null hypothesis when \(\alpha = 0.05\). And, in the second part of this example, we failed to reject the null hypothesis when \(\alpha = 0.01\). There must be some level of \(\alpha\), then, in which we cross the threshold from rejecting to not rejecting the null hypothesis. What is the smallest \(\alpha \text{ -level}\) that would still cause us to reject the null hypothesis?

We would, of course, reject any time the critical value was smaller than our test statistic −1.92:

That is, we would reject if the critical value were −1.645, −1.83, and −1.92. But, we wouldn't reject if the critical value were −1.93. The \(\alpha \text{ -level}\) associated with the test statistic −1.92 is called the P -value . It is the smallest \(\alpha \text{ -level}\) that would lead to rejection. In this case, the P -value is:

P ( Z < −1.92) = 0.0274

So far, all of the examples we've considered have involved a one-tailed hypothesis test in which the alternative hypothesis involved either a less than (<) or a greater than (>) sign. What happens if we weren't sure of the direction in which the proportion could deviate from the hypothesized null value? That is, what if the alternative hypothesis involved a not-equal sign (≠)? Let's take a look at an example.

What if we wanted to perform a " two-tailed " test? That is, what if we wanted to test:

\(H_0 \colon p = 0.90\) versus \(H_A \colon p \ne 0.90\)

at the \(\alpha = 0.05\) level?

Let's first consider the critical value approach . If we allow for the possibility that the sample proportion could either prove to be too large or too small, then we need to specify a threshold value, that is, a critical value, in each tail of the distribution. In this case, we divide the " significance level " \(\alpha\) by 2 to get \(\alpha/2\):

That is, our rejection rule is that we should reject the null hypothesis \(H_0 \text{ if } Z ≥ 1.96\) or we should reject the null hypothesis \(H_0 \text{ if } Z ≤ −1.96\). Alternatively, we can write that we should reject the null hypothesis \(H_0 \text{ if } |Z| ≥ 1.96\). Because our test statistic is −1.92, we just barely fail to reject the null hypothesis, because 1.92 < 1.96. In this case, we would say that there is insufficient evidence at the \(\alpha = 0.05\) level to conclude that the sample proportion differs significantly from 0.90.

Now for the P -value approach . Again, needing to allow for the possibility that the sample proportion is either too large or too small, we multiply the P -value we obtain for the one-tailed test by 2:

That is, the P -value is:

\(P=P(|Z|\geq 1.92)=P(Z>1.92 \text{ or } Z<-1.92)=2 \times 0.0274=0.055\)

Because the P -value 0.055 is (just barely) greater than the significance level \(\alpha = 0.05\), we barely fail to reject the null hypothesis. Again, we would say that there is insufficient evidence at the \(\alpha = 0.05\) level to conclude that the sample proportion differs significantly from 0.90.

Let's close this example by formalizing the definition of a P -value, as well as summarizing the P -value approach to conducting a hypothesis test.

The P -value is the smallest significance level \(\alpha\) that leads us to reject the null hypothesis.

Alternatively (and the way I prefer to think of P -values), the P -value is the probability that we'd observe a more extreme statistic than we did if the null hypothesis were true.

If the P -value is small, that is, if \(P ≤ \alpha\), then we reject the null hypothesis \(H_0\).

Note! Section  

By the way, to test \(H_0 \colon p = p_0\), some statisticians will use the test statistic:

\(Z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}}\)

rather than the one we've been using:

\(Z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}\)

One advantage of doing so is that the interpretation of the confidence interval — does it contain \(p_0\)? — is always consistent with the hypothesis test decision, as illustrated here:

For the sake of ease, let:

\(se(\hat{p})=\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\)

Two-tailed test. In this case, the critical region approach tells us to reject the null hypothesis \(H_0 \colon p = p_0\) against the alternative hypothesis \(H_A \colon p \ne p_0\):

if \(Z=\dfrac{\hat{p}-p_0}{se(\hat{p})} \geq z_{\alpha/2}\) or if \(Z=\dfrac{\hat{p}-p_0}{se(\hat{p})} \leq -z_{\alpha/2}\)

which is equivalent to rejecting the null hypothesis:

if \(\hat{p}-p_0 \geq z_{\alpha/2}se(\hat{p})\) or if \(\hat{p}-p_0 \leq -z_{\alpha/2}se(\hat{p})\)

if \(p_0 \geq \hat{p}+z_{\alpha/2}se(\hat{p})\) or if \(p_0 \leq \hat{p}-z_{\alpha/2}se(\hat{p})\)

That's the same as saying that we should reject the null hypothesis \(H_0 \text{ if } p_0\) is not in the \(\left(1-\alpha\right)100\%\) confidence interval!

Left-tailed test. In this case, the critical region approach tells us to reject the null hypothesis \(H_0 \colon p = p_0\) against the alternative hypothesis \(H_A \colon p < p_0\):

if \(Z=\dfrac{\hat{p}-p_0}{se(\hat{p})} \leq -z_{\alpha}\)

if \(\hat{p}-p_0 \leq -z_{\alpha}se(\hat{p})\)

if \(p_0 \geq \hat{p}+z_{\alpha}se(\hat{p})\)

That's the same as saying that we should reject the null hypothesis \(H_0 \text{ if } p_0\) is not in the upper \(\left(1-\alpha\right)100\%\) confidence interval:

\((0,\hat{p}+z_{\alpha}se(\hat{p}))\)

Value Hypothesis Fundamentals: A Complete Guide

Last updated on Fri Aug 23 2024

Imagine spending months or even years developing a new feature only to find out it doesn’t resonate with your users, argh! This kind of situation could be any worst Product manager’s nightmare.

There's a way to fix this problem called the Value Hypothesis . This idea helps builders to validate whether the ideas they’re working on are worth pursuing and useful to the people they want to sell to.

This guide will teach you what you need to know about Value Hypothesis and a step-by-step process on how to create a strong one. At the end of this post, you’ll learn how to create a product that satisfies your users.

Are you ready? Let’s get to it!

How a Value Hypothesis Helps Product Managers

Scrutinizing this hypothesis helps you as a developer to come up with a product that your customers like and love to use.

Product managers use the Value Hypothesis as a north star, ensuring focus on client needs and avoiding wasted resources. For more on this, read about the product management process .

Definition and Scope of Value Hypothesis

Let's get into the step-by-step process, but first, we need to understand the basics of the Value Hypothesis:

What Is a Value Hypothesis?

A Value Hypothesis is like a smart guess you can test to see if your product truly solves a problem for your customers. It’s your way of predicting how well your product will address a particular issue for the people you’re trying to help.

You need to know what a Value Hypothesis is, what it covers, and its key parts before you use it. To learn more about finding out what customers need, take a look at our guide on discovering features .

The Value Hypothesis does more than just help with the initial launch, it guides the whole development process. This keeps teams focused on what their users care about helping them choose features that their audience will like.

Critical Components of a Value Hypothesis

A strong Value Hypothesis rests on three key components:

Value Proposition: The Value Proposition spells out the main advantage your product gives to customers. It explains the "what" and "why" of your product showing how it eases a particular pain point.

This proposition targets a specific group of consumers. To learn more, check out our guide on roadmapping .

Customer Segmentation: Knowing and grasping your target audience is essential. This involves studying their demographics, needs, behaviors, and problems. By dividing your market, you can shape your value proposition to address the unique needs of each group.

Customer feedback surveys can prove priceless in this process. Find out more about this in our customer feedback surveys guide.

Problem Statement : The Problem Statement defines the exact issue your product aims to fix. It should zero in on a real fixable pain point your target users face. For hands-on applications, see our product launch communication plan .

Here are some key questions to guide you:

What are the primary challenges and obstacles faced by your target users?

What existing solutions are available, and where do they fall short?

What unmet needs or desires does your target audience have?

For a structured approach to prioritizing features based on customer needs, consider using a feature prioritization matrix .

Crafting a Strong Value Hypothesis

Now that we've covered the basics, let's look at how to build a convincing Value Hypothesis. Here's a two-step method, along with value hypothesis templates, to point you in the right direction:

1. Research and Analysis

To start with, you need to carry out market research. By carrying out proper market research, you will have an understanding of existing solutions and identify areas in which customers' needs are yet to be met. This is integral to effective idea tracking .

Next, use customer interviews, surveys, and support data to understand your target audience's problems and what they want. Check out our list of tools for getting customer feedback to help with this.

2. Finding Out What Customers Need

Once you've completed your research, it's crucial to identify your customers' needs. By merging insights from market research with direct user feedback, you can pinpoint the key requirements of your customers.

Here are some key questions to think about:

What are the most significant challenges that your target users encounter daily?

Which current solutions are available to them, and how do these solutions fail to fully address their needs?

What specific pain points are your target users struggling with that aren't being resolved?

Are there any gaps or shortcomings in the existing products or services that your customers use?

What unfulfilled needs or desires does your target audience express that aren't currently met by the market?

To prioritize features based on customer needs in a structured way, think about using a feature prioritization matrix .

Validating the Value Hypothesis

Once you've created your Value Hypothesis with a template, you need to check if it holds up. Here's how you can do this:

MVP Testing

Build a minimum viable product (MVP)—a basic version of your product with essential functions. This lets you test your value proposition with actual users and get feedback without spending too much. To achieve the best outcomes, look into the best practices for customer feedback software .

Prototyping

Build mock-ups to show your product idea. Use these mock-ups to get user input on the user experience and overall value offer.

Metrics for Evaluation

After you've gathered data about your hypothesis, it's time to examine it. Here are some metrics you can use:

User Engagement : Monitor stats like time on the platform, feature use, and return visits to see how much users interact with your MVP or mock-up.

Conversion Rates : Check conversion rates for key actions like sign-ups, buys, or feature adoption. These numbers help you judge if your value offer clicks with users. To learn more, read our article on SaaS growth benchmarks .

Iterative Improvement of Value Hypothesis

The Value Hypothesis framework shines because you can keep making it better. Here's how to fine-tune your hypothesis:

Set up an ongoing system to gather user data as you develop your product.

Look at what users say to spot areas that need work then update your value proposition based on what you learn.

Read about managing product updates to keep your hypotheses current.

Adaptation to Market Changes

The market keeps changing, and your Value Hypothesis should too. Stay up to date on what's happening in your industry and watch how users' habits change. Tweak your value proposition to stay useful and ahead of the competition.

Here are some ways to keep your Value Hypothesis fresh:

Do market research often to keep up with what's happening in your industry and what your competitors are up to.

Keep an eye on what users are saying to spot new problems or things they need but don't have yet.

Try out different value statements and features to see which ones your audience likes best.

To keep your guesses up-to-date, check out our guide on handling product changes .

Common Mistakes to Avoid

While the Value Hypothesis approach is powerful, it's key to steer clear of these common traps:

Avoid Confirmation Bias : People tend to focus on data that backs up their initial guesses. But it's key to look at feedback that goes against your ideas and stay open to different views.

Watch out for Shiny Object Syndrome : Don't let the newest fads sway you unless they solve a main customer problem. Your value proposition should fix actual issues for your users.

Don't Cling to Your First Hypothesis : As the market changes, your value proposition should too. Be ready to shift your hypothesis when new evidence and user feedback comes in.

Don't Mix Up Busywork with Real Progress : Getting user feedback is key, but making sense of it brings real value. Look at the data to find useful insights that can shape your product. To learn more about this, check out our guide on handling customer feedback .

Value Hypothesis: Action Points

To build a product that succeeds, you need to know your target users inside out and understand how you help them. The Value Hypothesis framework gives you a step-by-step way to do this.

If you follow the steps in this guide, you can create a strong value proposition, check if it works, and keep improving it to ensure your product stays useful and important to your customers.

Keep in mind, a good Value Hypothesis changes as your product and market change. When you use data and put customers first, you're on the right track to create a product that works.

Want to put the Value Hypothesis framework into action? Check out our top templates for creating product roadmaps to streamline your process. Think about using featureOS to manage customer feedback. This tool makes it easier to collect, examine, and put user feedback to work.

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