what is a hypothesis test rejection region

Upper-tailed, Lower-tailed, Two-tailed Tests

The research or alternative hypothesis can take one of three forms. An investigator might believe that the parameter has increased, decreased or changed. For example, an investigator might hypothesize:  

The exact form of the research hypothesis depends on the investigator's belief about the parameter of interest and whether it has possibly increased, decreased or is different from the null value. The research hypothesis is set up by the investigator before any data are collected.

Rejection Region for Upper-Tailed Z Test (H : μ > μ ) with α=0.05

The decision rule is: Reject H if Z 1.645.

Rejection Region for Lower-Tailed Z Test (H 1 : μ < μ 0 ) with α =0.05

The decision rule is: Reject H 0 if Z < 1.645.

Rejection Region for Two-Tailed Z Test (H 1 : μ ≠ μ 0 ) with α =0.05

The decision rule is: Reject H 0 if Z < -1.960 or if Z > 1.960.

The complete table of critical values of Z for upper, lower and two-tailed tests can be found in the table of Z values to the right in "Other Resources."

Critical values of t for upper, lower and two-tailed tests can be found in the table of t values in "Other Resources."

• Step 4. Compute the test statistic.  

Here we compute the test statistic by substituting the observed sample data into the test statistic identified in Step 2.

• Step 5. Conclusion.  

The final conclusion is made by comparing the test statistic (which is a summary of the information observed in the sample) to the decision rule. The final conclusion will be either to reject the null hypothesis (because the sample data are very unlikely if the null hypothesis is true) or not to reject the null hypothesis (because the sample data are not very unlikely).  

If the null hypothesis is rejected, then an exact significance level is computed to describe the likelihood of observing the sample data assuming that the null hypothesis is true. The exact level of significance is called the p-value and it will be less than the chosen level of significance if we reject H 0 .

Statistical computing packages provide exact p-values as part of their standard output for hypothesis tests. In fact, when using a statistical computing package, the steps outlined about can be abbreviated. The hypotheses (step 1) should always be set up in advance of any analysis and the significance criterion should also be determined (e.g., α =0.05). Statistical computing packages will produce the test statistic (usually reporting the test statistic as t) and a p-value. The investigator can then determine statistical significance using the following: If p < α then reject H 0 .  

• Step 1. Set up hypotheses and determine level of significance

H 0 : μ = 191 H 1 : μ > 191                 α =0.05

The research hypothesis is that weights have increased, and therefore an upper tailed test is used.

• Step 2. Select the appropriate test statistic.

Because the sample size is large (n > 30) the appropriate test statistic is

• Step 3. Set up decision rule.  

In this example, we are performing an upper tailed test (H 1 : μ> 191), with a Z test statistic and selected α =0.05.   Reject H 0 if Z > 1.645.

We now substitute the sample data into the formula for the test statistic identified in Step 2.  

We reject H 0 because 2.38 > 1.645. We have statistically significant evidence at a =0.05, to show that the mean weight in men in 2006 is more than 191 pounds. Because we rejected the null hypothesis, we now approximate the p-value which is the likelihood of observing the sample data if the null hypothesis is true. An alternative definition of the p-value is the smallest level of significance where we can still reject H 0 . In this example, we observed Z=2.38 and for α=0.05, the critical value was 1.645. Because 2.38 exceeded 1.645 we rejected H 0 . In our conclusion we reported a statistically significant increase in mean weight at a 5% level of significance. Using the table of critical values for upper tailed tests, we can approximate the p-value. If we select α=0.025, the critical value is 1.96, and we still reject H 0 because 2.38 > 1.960. If we select α=0.010 the critical value is 2.326, and we still reject H 0 because 2.38 > 2.326. However, if we select α=0.005, the critical value is 2.576, and we cannot reject H 0 because 2.38 < 2.576. Therefore, the smallest α where we still reject H 0 is 0.010. This is the p-value. A statistical computing package would produce a more precise p-value which would be in between 0.005 and 0.010. Here we are approximating the p-value and would report p < 0.010.                  

In all tests of hypothesis, there are two types of errors that can be committed. The first is called a Type I error and refers to the situation where we incorrectly reject H 0 when in fact it is true. This is also called a false positive result (as we incorrectly conclude that the research hypothesis is true when in fact it is not). When we run a test of hypothesis and decide to reject H 0 (e.g., because the test statistic exceeds the critical value in an upper tailed test) then either we make a correct decision because the research hypothesis is true or we commit a Type I error. The different conclusions are summarized in the table below. Note that we will never know whether the null hypothesis is really true or false (i.e., we will never know which row of the following table reflects reality).

Table - Conclusions in Test of Hypothesis

In the first step of the hypothesis test, we select a level of significance, α, and α= P(Type I error). Because we purposely select a small value for α, we control the probability of committing a Type I error. For example, if we select α=0.05, and our test tells us to reject H 0 , then there is a 5% probability that we commit a Type I error. Most investigators are very comfortable with this and are confident when rejecting H 0 that the research hypothesis is true (as it is the more likely scenario when we reject H 0 ).

When we run a test of hypothesis and decide not to reject H 0 (e.g., because the test statistic is below the critical value in an upper tailed test) then either we make a correct decision because the null hypothesis is true or we commit a Type II error. Beta (β) represents the probability of a Type II error and is defined as follows: β=P(Type II error) = P(Do not Reject H 0 | H 0 is false). Unfortunately, we cannot choose β to be small (e.g., 0.05) to control the probability of committing a Type II error because β depends on several factors including the sample size, α, and the research hypothesis. When we do not reject H 0 , it may be very likely that we are committing a Type II error (i.e., failing to reject H 0 when in fact it is false). Therefore, when tests are run and the null hypothesis is not rejected we often make a weak concluding statement allowing for the possibility that we might be committing a Type II error. If we do not reject H 0 , we conclude that we do not have significant evidence to show that H 1 is true. We do not conclude that H 0 is true.

 The most common reason for a Type II error is a small sample size.

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Content ©2017. All Rights Reserved. Date last modified: November 6, 2017. Wayne W. LaMorte, MD, PhD, MPH

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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

Acceptance Region: Simple Definition & Example

Hypothesis Testing >

More Formal Definition of Acceptance Region

According to the The Concise Encyclopedia of Statistics , the acceptance region is “…the interval within the sampling distribution of the test statistic that is consistent with the null hypothesis H 0 from hypothesis testing .” In more simple terms, let’s say you run a hypothesis test like a z-test . The results of the test come in the form of a z-value , which has a large range of possible values. Within that range of values, some will fall into an interval that suggests the null hypothesis is correct. That interval is the acceptance region.

Why is it “Provisionally” Acceptance?

You provisionally accept the null hypothesis because a hypothesis test doesn’t tell you which hypothesis is true (the null or alternate hypothesis ), or even which is probably true. The only thing is tests is whether there’s enough evidence in your data to reject the null hypothesis. Failure to accept the alternate hypothesis doesn’t make the null hypothesis true.

Let’s say our “experiment” is where you caught a child red-handed with a stolen cookie:

• Null hypothesis (H 0 ): The child didn’t steal the cookie (innocent until proven guilty!).
• Alternate hypothesis (H 1 ): The child did steal the cookie.

You’re pretty certain the child stole the cookie. But after gathering all evidence, you don’t find enough evidence to say for sure that the child is guilty. Therefore, there isn’t enough evidence in support of the alternate hypothesis that the child is guilty . In other words, you can’t reject the null hypothesis that the child is innocent in favor of the hypothesis that the child is guilty. That doesn’t mean the child is innocent. You just didn’t have enough evidence to prove them guilty. Although your result fell into the acceptance region, you don’t actually “accept” the null hypothesis of innocence. You just provisionally (perhaps begrudgingly) accept it and let the child off without punishment. Later on you might find crumbs in their bed, leading you to revisit your findings.

This subtle difference may seem pedantic, and in elementary statistics it’s usually not an important matter to stress. However, if you plan to publish your results you should never say you “accept the null hypothesis”. You can say that you provisionally accept it, or that you failed to reject it . Check with your intended publication (or with your professor) to see what wording they prefer.

Dodge, Y. (2008). The Concise Encyclopedia of Statistics . Springer. Gonick, L. (1993). The Cartoon Guide to Statistics . HarperPerennial.

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6a.4.2 - more on the p-value and rejection region approach, two methods for making a statistical decision section  .

Of the two methods for making a statistical decision, the p-value approach is more commonly used and provided in published literature. However, understanding the rejection region approach can go a long way in one's understanding of the p-value method. In the video, we show how the two methods are related. Regardless of the method applied, the conclusions from the two approaches are exactly the same.

Video: The Rejection Region vs the P-Value Approach

Comparing the Two Approaches Section  

Both approaches will ensure the same conclusion and either one will work. However, using the p-value approach has the following advantages:

• Using the rejection region approach, you need to check the table or software for the critical value every time you use a different \(\alpha \) value.
• In addition to just using it to reject or not reject \(H_0 \) by comparing p-value to \(\alpha \) value, the p-value also gives us some idea of the strength of the evidence against \(H_0 \).

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Different ways of determining rejection region of a two sided test

In a two sided test, assume the test statistic has been chosen to be $T(X)$ and the distribution of $T(X)$ under null hypothesis is also known to be $F$. Let the significance level be $\alpha$.

I can come up with two different ways to determine the rejection region:

$\{|T(x) - \mu| > c\}$. $\mu$ is the mean of the distribution $F$ of $T(X)$under null, and $c$ is determined by solving $$\inf_{c \geq 0} c$$ subject to $$P_{T(X) \sim F} (|T(X) - \mu| > c) \leq \alpha.$$ So the rejection region is symmetric around $\mu$.

$\{T(x) > c_1\} \cup \{T(x) < c_2\}$. $c_1$ and $c_2$ are determined by solving $$\inf_{c_1 \in \mathbb R} c_1$$ subject to $$P_{T(X) \sim F} (T(X) > c_1) \leq \alpha/2$$ and $$\sup_{c_2 \in \mathbb R} c_2$$ subject to $$P_{T(X) \sim F} (T(X) < c_2) \leq \alpha/2.$$ So the rejection region evenly split $\alpha$ to both sides.

Am I correct that those two methods will agree when the null distribution $F$of $T(X)$ is symmetric around its mean $\mu$, and may not agree when $F$ isn't symmetric around $\mu$?

I was wondering what advantage and disadvantages these two methods have? Which one is recommended and when?

Are both methods used in some textbooks? If yes, references?

What are some other methods for two-sided tests? For example, can we generalized the second method by splitting $\alpha$ arbitrarily unevenly to the two sides?

Consider the relation between rejection region in testing and confidence interval. Are the above discussions also apply to confidence intervals?

Thanks and regards!

• hypothesis-testing

Question 1. Right, if $F$ is symmetric (and continuous: weird things may happen if some points have non-zero weight and you cut your interval right on them) you have $P(T(X)-\mu) > c = P(T(X)-\mu < -c)$, so you can split the rejection region into two unbounded intervals, each of probability $\alpha / 2$.

Questions 2,4. For any set $C$, as long as $P(T(X) \in C) = \alpha$, you're fine. You could even choose a region such as $[\mu-d,\mu + d]$ ! It is however not natural, because one would like your rejection set to include weird values of the statistic (values that are far from $\mu$) rather than normal values. Why? Because the test statistic is expected to be a measure of the distance between the data and the null hypothesis. For instance, in the lady tasting tea, the test statistic is the number of correctly classified cups. One would expect that the higher it is, the more the ability of the lady is proven. A weird rejection set could be designed, that could include the lady correctly classifying all cups, but not the lady correctly classifying all cups but one. It would be perfectly acceptable from a formal point of vue.

That insisting on having centered confidence intervals is not based on mathematics strictly can be seen easily: if $[a,b]$ is a centered confidence interval for parameter $\theta$, $[f(a),f(b)]$ has no reason to be a centered confidence interval for transformed parameter $f(\theta)$. For instance, say your data is sound power, and that you want to relate that to medical damages or sound perception; do you want intervals centered on the original scale ($W.m^{-2}$ for instance), or on the transformed log scale (dB) ?

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