Textbook Question
n Exercises 1–6, the statement represents a claim. Write its complement and state which is H0 and which is Ha.
μ ≤ 375
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9. Hypothesis Testing for One Sample
Steps in Hypothesis Testing
2:41 minutesProblem 7.5.11
Textbook Question
In Exercises 7–12, find the critical value(s) and rejection region(s) for the type of chi-square test with sample size n and level of significance α.
Two-tailed test, n=81,α=0.10
Verified step by step guidance1
Determine the degrees of freedom (df) for the chi-square test. The formula for degrees of freedom is df = n - 1, where n is the sample size. In this case, n = 81, so df = 81 - 1.
Identify the level of significance (α) for the test. For a two-tailed test, the significance level is split equally between the two tails of the chi-square distribution. Thus, each tail will have an area of α/2 = 0.10/2.
Use a chi-square distribution table or statistical software to find the critical values corresponding to the upper and lower tails of the distribution. Look up the chi-square values for df = 80 and cumulative probabilities of 1 - α/2 (upper tail) and α/2 (lower tail).
Define the rejection regions based on the critical values. The rejection region for a two-tailed test includes chi-square values less than the lower critical value and greater than the upper critical value.
Summarize the critical values and rejection regions. Clearly state the critical values and the intervals that define the rejection regions for the test.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Chi-Square Test
The chi-square test is a statistical method used to determine if there is a significant association between categorical variables. It compares the observed frequencies in each category to the frequencies expected under the null hypothesis. This test is commonly used in hypothesis testing to assess goodness-of-fit or independence.
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Critical Value
A critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the chosen significance level (α) and the distribution of the test statistic. For a chi-square test, critical values can be found using chi-square distribution tables based on the degrees of freedom and the specified α.
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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 chi-square test with a significance level of α, the rejection regions are determined by the critical values, indicating where the test statistic must fall to reject the null hypothesis.
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