Textbook Question
a. Determine the critical value for a right-tailed test of a population mean at the α = 0.01 level of significance with 22 degrees of freedom.
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9. Hypothesis Testing for One Sample
Critical Values and Rejection Regions
2:35 minutesProblem 7.4.5
Textbook Question
In Exercises 3–8, find the critical value(s) and rejection region(s) for the type of t-test with level of significance alpha and sample size n.
Right-tailed test, α=0.05, n=23
Verified step by step guidance1
Step 1: Identify the degrees of freedom (df) for the t-test. The degrees of freedom are calculated as df = n - 1, where n is the sample size. For this problem, df = 23 - 1.
Step 2: Recognize that this is a right-tailed t-test. This means the rejection region will be in the upper tail of the t-distribution.
Step 3: Use the level of significance (α = 0.05) and the degrees of freedom (df = 22) to find the critical value from a t-distribution table or statistical software. Look up the t-value corresponding to a cumulative probability of 1 - α = 0.95 for df = 22.
Step 4: Define the rejection region. For a right-tailed test, the rejection region consists of all t-values greater than the critical value found in Step 3.
Step 5: Summarize the results. The critical value and rejection region are now determined, and these will be used to decide whether to reject the null hypothesis in the context of the t-test.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Critical Value
A critical value is a point on the scale of the test statistic that separates the region where the null hypothesis is rejected from the region where it is not rejected. In hypothesis testing, critical values are determined based on the significance level (alpha) and the distribution of the test statistic. For a right-tailed t-test, the critical value corresponds to the point where the cumulative probability equals 1 - alpha.
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Rejection Region
The rejection region is the range of values for the test statistic that leads to the rejection of the null hypothesis. In a right-tailed test, this region is located to the right of the critical value. If the calculated test statistic falls within this region, it indicates that the observed data is sufficiently unlikely under the null hypothesis, prompting its rejection.
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09:56Step 4: State Conclusion
T-Test
A t-test is a statistical test used to determine if there is a significant difference between the means of two groups, or between a sample mean and a known value. It is particularly useful when the sample size is small (typically n < 30) and the population standard deviation is unknown. The t-test uses the t-distribution, which accounts for the additional uncertainty introduced by estimating the population standard deviation from the sample.
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