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. Show
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: 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 TestsTwo-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 TestsOne-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 IntervalsConfidence 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.
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 ValueUnfortunately, 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.
Related post: Degrees of Freedom Critical Value CalculatorAnother 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:
The calculator will display the same ±1.96 values we found earlier in this article.
Contents: What does it mean to reject the null hypothesis?Watch the video for an example: Support or Reject a Null Hypothesis P Value Method Watch this video on YouTube. Can’t see the video? Click here. In many statistical tests, you’ll want to either reject or support the null hypothesis. For elementary statistics students, the term can be a tricky term to grasp, partly because the name “null hypothesis” doesn’t make it clear about what the null hypothesis actually is! OverviewThe null hypothesis can be thought of as a nullifiable hypothesis. That means you can nullify it, or reject it. What happens if you reject the null hypothesis? It gets replaced with the alternate hypothesis, which is what you think might actually be true about a situation. For example, let’s say you think that a certain drug might be responsible for a spate of recent heart attacks. The drug company thinks the drug is safe. The null hypothesis is always the accepted hypothesis; in this example, the drug is on the market, people are using it, and it’s generally accepted to be safe. Therefore, the null hypothesis is that the drug is safe. The alternate hypothesis — the one you want to replace the null hypothesis, is that the drug isn’t safe. Rejecting the null hypothesis in this case means that you will have to prove that the drug is not safe. Vioxx was pulled from the market after it was linked to heart problems. To reject the null hypothesis, perform the following steps:Step 1: State the null hypothesis. When you state the null hypothesis, you also have to state the alternate hypothesis. Sometimes it is easier to state the alternate hypothesis first, because that’s the researcher’s thoughts about the experiment. How to state the null hypothesis (opens in a new window). Step 2: Support or reject the null hypothesis. Several methods exist, depending on what kind of sample data you have. For example, you can use the P-value method. For a rundown on all methods, see: Support or reject the null hypothesis. If you are able to reject the null hypothesis in Step 2, you can replace it with the alternate hypothesis. That’s it! When to Reject the Null hypothesisBasically, you reject the null hypothesis when your test value falls into the rejection region. There are four main ways you’ll compute test values and either support or reject your null hypothesis. Which method you choose depends mainly on if you have a proportion or a p-value. Support or Reject the Null Hypothesis: StepsClick the link the skip to the situation you need to support or reject null hypothesis for: Support or Reject Null Hypothesis with a P ValueIf you have a P-value, or are asked to find a p-value, follow these instructions to support or reject the null hypothesis. This method works if you are given an alpha level and if you are not given an alpha level. If you are given a confidence level, just subtract from 1 to get the alpha level. See: How to calculate an alpha level. Step 1: State the null hypothesis and the alternate hypothesis (“the claim”). Step 2: Find the critical value. We’re dealing with a normally distributed population, so the critical value is a z-score. Click here if you want easy, step-by-step instructions for solving this formula. Step 4: Find the P-Value by looking up your answer from step 3 in the z-table. To get the p-value, subtract the area from 1. For example, if your area is .990 then your p-value is 1-.9950 = 0.005. Note: for a two-tailed test, you’ll need to halve this amount to get the p-value in one tail. Step 5: Compare your answer from step 4 with the α value given in the question. Should you support or reject the null hypothesis? P-Value GuidelinesUse these general guidelines to decide if you should reject or keep the null: If p value > .10 → “not significant” If p value ≤ .10 → “marginally significant” If p value ≤ .05 → “significant” If p value ≤ .01 → “highly significant.” That’s it! Back to Top Support or Reject Null Hypothesis for a ProportionSometimes, you’ll be given a proportion of the population or a percentage and asked to support or reject null hypothesis. In this case you can’t compute a test value by calculating a z-score (you need actual numbers for that), so we use a slightly different technique. Watch the video for an example: Hypothesis Test for a Proportion Watch this video on YouTube. Can’t see the video? Click here. Example question: A researcher claims that Democrats will win the next election. 4300 voters were polled; 2200 said they would vote Democrat. Decide if you should support or reject null hypothesis. Is there enough evidence at α=0.05 to support this claim? Step 1: State the null hypothesis and the alternate hypothesis (“the claim”). Step 2: Compute 2200/4300 = 0.512. Step 3: Use the following formula to calculate your test value. Where: Phat is calculated in Step 2 P the null hypothesis p value (.05) Q is 1 – p The z-score is: Step 4: Look up Step 3 in the z-table to get .9418. Step 5: Calculate your p-value by subtracting Step 4 from 1. Step 6: Compare your answer from step 5 with the α value given in the question. Support or reject the null hypothesis? If step 5 is less than α, reject the null hypothesis, otherwise do not reject it. In this case, .582 (5.82%) is not less than our α, so we do not reject the null hypothesis. Back to Top Support or Reject Null Hypothesis for a Proportion: Second exampleExample question: A researcher claims that more than 23% of community members go to church regularly. In a recent survey, 126 out of 420 people stated they went to church regularly. Is there enough evidence at α = 0.05 to support this claim? Use the P-Value method to support or reject null hypothesis. Step 1: State the null hypothesis and the alternate hypothesis (“the claim”). Ho:p ≤ 0.23; H1:p > 0.23 (claim) Step 2: Compute by dividing the number of positive respondents from the number in the random sample: Step 3: Find ‘p’ by converting the stated claim to a decimal: 23% = 0.23. Also, find ‘q’ by subtracting ‘p’ from 1: 1 – 0.23 = 0.77. Step 4: Use the following formula to calculate your test value. Click here if you want easy, step-by-step instructions for solving this formula. If formulas confuse you, this is asking you to:
Step 5: Find the P-Value by looking up your answer from step 5 in the z-table. The z-score for 3.41 is .4997. Subtract from 0.500: 0.500-.4997 = 0.003. Step 6: Compare your P-value to α. Support or reject null hypothesis? If the P-value is less, reject the null hypothesis. If the P-value is more, keep the null hypothesis. Note: In Step 5, I’m using the z-table on this site to solve this problem. Most textbooks have the right of z-table. If you’re seeing .9997 as an answer in your textbook table, then your textbook has a “whole z” table, in which case don’t subtract from .5, subtract from 1. 1-.9997 = 0.003. Back to Top Check out our Youtube channel for video tips! ReferencesEveritt, B. S.; Skrondal, A. (2010), The Cambridge Dictionary of Statistics, Cambridge University Press.
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When to reject null hypothesis critical value
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