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 7.31
In Exercises 29–34, find the critical value(s) and rejection region(s) for the type of t-test with level of significance α and sample size n.
Right-tailed test, α=0.02, n=63
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Determine the degrees of freedom (df) for the t-test. The formula for degrees of freedom is df = n - 1, where n is the sample size. In this case, df = 63 - 1.
Identify the level of significance (α) for the test. Here, α = 0.02, which corresponds to the probability of rejecting the null hypothesis when it is true.
Since this is a right-tailed test, the critical value corresponds to the t-score where the area to the right under the t-distribution curve equals α. Use a t-distribution table or statistical software to find the t-score for df = 62 and α = 0.02.
Define the rejection region. For a right-tailed test, the rejection region consists of all t-scores greater than the critical value found in the previous step.
Summarize the results: The critical value and rejection region are determined based on the t-distribution table or software output. The rejection region is t > critical value.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Critical Value
The 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 t-test, the critical value is the point beyond which the null hypothesis is rejected, indicating that the observed data is statistically significant.
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Critical Values: t-Distribution
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 right-tailed t-test, this region is located to the right of the critical value on the t-distribution curve. If the calculated test statistic falls within this region, it suggests that the sample provides sufficient evidence to reject the null hypothesis at the specified significance level.
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Guided course
09:56Step 4: State Conclusion
T-Test
A t-test is a statistical test used to compare the means of two groups or to compare a sample mean to a known value when the population standard deviation is unknown. It is particularly useful for small sample sizes (typically n < 30) and is based on the t-distribution. The type of t-test (one-sample, independent two-sample, or paired sample) depends on the data structure and research question.
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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
Graphical Analysis In Exercises 17–20, match the alternative hypothesis with its graph. Then state the null hypothesis and sketch its graph.
Ha: μ ≠ 3
a.
b.
c.
d.
Textbook Question
Hypothesis Testing Using a P-Value In Exercises 33–38,
a. identify the claim and state and .
b. find the standardized test statistic z.
c. find the corresponding P-value.
d. decide whether to reject or fail to reject the null hypothesis.
e. interpret the decision in the context of the original claim.
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Textbook Question
Faculty Classroom Hours The dean of a university estimates that the mean number of classroom hours per week for full-time faculty is 11.0. As a member of the student council, you want to test this claim. A random sample of the number of classroom hours for eight full-time faculty for one week is shown in the table at the left. At α=0.01, can you reject the dean’s claim?
Textbook Question
True or False? In Exercises 5–10, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
A statistical hypothesis is a statement about a sample.