Textbook Question
a. Determine the critical value for a right-tailed test of a population standard deviation with 18 degrees of freedom at the α = 0.05 level of significance.
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9. Hypothesis Testing for One Sample
Critical Values and Rejection Regions
3:08 minutesProblem 7.4.7
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.
Two-tailed test, α=0.05, n=27
Verified step by step guidance1
Determine the degrees of freedom (df) for the t-test. The formula for degrees of freedom in a one-sample t-test is df = n - 1, where n is the sample size. Substitute n = 27 into the formula to calculate df.
Identify the level of significance (α) for the test. In this case, α = 0.05. Since it is a two-tailed test, divide α by 2 to account for both tails of the distribution. This gives α/2 = 0.025 for each tail.
Use a t-distribution table or statistical software to find the critical t-value corresponding to df = 26 (calculated in step 1) and α/2 = 0.025. The critical t-value is the value where the cumulative probability in the tail equals 0.025.
Define the rejection regions for the two-tailed test. The rejection regions are the areas in the tails of the t-distribution where the test statistic falls beyond the critical t-values. For a two-tailed test, the rejection regions are t < -t_critical and t > t_critical.
Summarize the critical values and rejection regions. State the critical t-values (positive and negative) and the corresponding rejection regions based on the results from the t-distribution table or software.
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Video duration:
3mKey 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 a two-tailed test, critical values are determined based on the significance level (alpha) and the degrees of freedom, which is calculated as n-1 for a t-test. For α=0.05 and n=27, the critical values help define the rejection regions.
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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 two-tailed test, this region is split between both tails of the distribution. For a significance level of α=0.05, the rejection regions are located in the extreme ends of the t-distribution, beyond the critical values determined for the test.
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Two-Tailed Test
A two-tailed test is a statistical test that evaluates whether a sample mean is significantly different from a population mean in either direction (higher or lower). This type of test is appropriate when the alternative hypothesis does not specify a direction of the effect. In this case, with α=0.05, the test assesses the likelihood of observing a sample mean that is either significantly greater than or less than the hypothesized population mean.
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