Textbook Question
In Exercises 13–18, test the claim about the population mean μ at the level of significance α. Assume the population is normally distributed.
Claim: μ≠52,200; α=0.05. Sample statistics: x_bar=53,220, s=2700, n=34
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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 7.2.28
Finding Critical Values and Rejection Regions In Exercises 23–28, find the critical value(s) and rejection region(s) for the type of z-test with level of significance α. Include a graph with your answer.
Two-tailed test, α = 0.12
Verified step by step guidance1
Step 1: Understand the problem. This is a two-tailed z-test with a significance level (α) of 0.12. In a two-tailed test, the rejection regions are located in both tails of the standard normal distribution, and the total area of the rejection regions equals α.
Step 2: Divide the significance level (α) equally between the two tails. Since α = 0.12, each tail will have an area of α/2 = 0.12/2 = 0.06.
Step 3: Use the standard normal distribution table (z-table) or a statistical calculator to find the z-scores that correspond to the cumulative probabilities of 0.06 (left tail) and 1 - 0.06 = 0.94 (right tail). These z-scores are the critical values.
Step 4: Define the rejection regions. The rejection regions are the areas in the tails of the distribution where the test statistic falls outside the critical values. For a two-tailed test, the rejection regions are z < -z_critical (left tail) and z > z_critical (right tail).
Step 5: Visualize the result. Draw a standard normal distribution curve, mark the critical values on the z-axis, and shade the rejection regions in both tails. This graph helps to clearly illustrate the critical values and rejection regions.
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Key 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 hypothesis testing, critical values are determined based on the significance level (α) and the type of test (one-tailed or two-tailed). For a two-tailed test, critical values are found at both extremes of the distribution.
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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 divided into two parts, corresponding to the critical values on either side of the distribution. The size of the rejection region is determined by the significance level (α), which indicates the probability of making a Type I error.
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09:56Step 4: State Conclusion
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 (greater than or less than). This type of test is used when the alternative hypothesis does not specify a direction of the effect. For a significance level of α = 0.12, the critical values will be located at the 6% and 94% percentiles of the standard normal distribution.
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Related Practice
Textbook Question
In Exercises 15–22, test the claim about the population variance or standard deviation at the level of significance Assume the population is normally distributed.
Claim: σ^2=63, α=0.01 . Sample statistics: s^2=58, n=29
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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
Getting at the Concept Explain why a level of significance of α=0 is not used.
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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
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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.
The level of significance is the maximum probability you allow for rejecting a null hypothesis when it is actually true.
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