Textbook Question
b. Determine the critical value for a left-tailed test of a population mean at the α = 0.01 level of significance based on a sample size of n = 40.
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9. Hypothesis Testing for One Sample
Critical Values and Rejection Regions
2:18 minutesProblem 7.RE.54
Textbook Question
In Exercises 51–54, find the critical value(s) and rejection region(s) for the type of chi-square test with sample size n and level of significance.
Left-tailed test, n=6, α=0.05
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 = 6, so df = 6 - 1 = 5.
Identify the level of significance (α) for the test. Here, α = 0.05, which represents the probability of rejecting the null hypothesis when it is true.
Since this is a left-tailed test, locate the critical value for the chi-square distribution corresponding to df = 5 and α = 0.05. Use a chi-square distribution table or statistical software to find this value.
Define the rejection region for the left-tailed test. The rejection region consists of all chi-square values less than the critical value obtained in the previous step.
Summarize the critical value and rejection region. State that if the test statistic falls within the rejection region, the null hypothesis will be rejected at the 0.05 significance level.
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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 particularly useful in analyzing contingency tables and goodness-of-fit problems.
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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 left-tailed chi-square test, the critical value indicates the point below which the test statistic must fall to reject the null hypothesis.
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Rejection Region
The rejection region is the set of values for the test statistic that leads to the rejection of the null hypothesis. In a left-tailed test, this region is located to the left of the critical value on the chi-square distribution. If the calculated test statistic falls within this region, it suggests that the observed data is significantly different from what is expected under the null hypothesis.
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