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Ch. 7 - Hypothesis Testing with One Sample
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 7 - Hypothesis Testing with One SampleProblem 7.5.16
Chapter 7, Problem 7.5.16

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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Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: σ² = 63, and the alternative hypothesis is H₁: σ² ≠ 63 (since this is a two-tailed test).
Step 2: Determine the test statistic for a chi-square test of variance. The formula is χ² = ((n - 1) * s²) / σ², where n is the sample size, s² is the sample variance, and σ² is the claimed population variance.
Step 3: Calculate the degrees of freedom (df), which is given by df = n - 1. In this case, df = 29 - 1 = 28.
Step 4: Determine the critical values for the chi-square distribution at the significance level α = 0.01 for a two-tailed test. Use a chi-square table or statistical software to find the critical values for df = 28 and α/2 = 0.005 in each tail.
Step 5: Compare the calculated test statistic to the critical values. If the test statistic falls outside the range defined by the critical values, reject the null hypothesis H₀. Otherwise, fail to reject H₀. Interpret the result in the context of the problem.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Population Variance

Population variance (σ²) measures the dispersion of a set of values in a population. It is calculated as the average of the squared differences from the mean. In hypothesis testing, we often compare the sample variance (s²) to the claimed population variance to determine if there is enough evidence to reject the null hypothesis.
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Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample statistics to determine whether to reject H0 at a specified significance level (α). In this case, we are testing if the population variance is equal to a specific value.
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Step 1: Write Hypotheses

Significance Level (α)

The significance level (α) is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. It represents the threshold for determining whether the observed data is statistically significant. In this scenario, α is set at 0.01, indicating a 1% risk of concluding that a difference exists when there is none.
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Related Practice
Textbook Question

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

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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.


Left-tailed test, n=7,α=0.01

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Textbook Question

Hypothesis Testing Using Rejection Regions In Exercises 19–26, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic t, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the population is normally distributed.


Annual Salary An employment information service claims the mean annual salary for senior level statisticians is more than \(124,000. The annual salaries (in dollars) for a random sample of 12 senior level statisticians are shown in the table at the left. At α=0.01, is there enough evidence to support the claim that the mean salary is more than \)124,000?


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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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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.


Paying for College According to a recent survey, 54% of today’s college students used student loans to pay for college.

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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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