Textbook Question
Explain how to find critical values for a t-distribution.
Ch. 7 - Hypothesis Testing with One Sample
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 7, Problem 7.5.7
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.
Right-tailed test, n=27,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, df = 27 - 1.
Identify the level of significance (α). For this problem, α = 0.05, which represents the probability of rejecting the null hypothesis when it is true.
Since this is a right-tailed test, locate the critical value of the chi-square distribution corresponding to df = 26 and α = 0.05. Use a chi-square distribution table or statistical software to find this value.
Define the rejection region. For a right-tailed test, the rejection region consists of all chi-square values greater than the critical value obtained in the previous step.
Summarize the results: The critical value and rejection region are now determined. The rejection region is where the test statistic exceeds the critical value, indicating evidence to reject the null hypothesis.
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Key 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 right-tailed chi-square test, the critical value indicates the point beyond which the test statistic is considered significant.
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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 suggests that the observed data is unlikely under the null hypothesis, prompting its rejection.
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Related Practice
Textbook Question
Identifying a Test In Exercises 21–24, determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed.
Ha: μ ≥ 5.2
H0: μ < 5.2
Textbook Question
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Ha: μ ≠ 3
a.
b.
c.
d.
Textbook Question
Stating the Null and Alternative Hypotheses In Exercises 25–30, write the claim as a mathematical statement. State the null and alternative hypotheses, and identify which represents the claim.
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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.
Left-tailed test, α=0.01, n=35
Textbook Question
Identifying Type I and Type II Errors In Exercises 31–36, describe type I and type II errors for a hypothesis test of the indicated claim.
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