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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 7.5.30
Chapter 7, Problem 7.5.30

Hypothesis Testing Using Rejection Regions In Exercises 23–30, (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 X^2, (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.


Salaries The annual salaries (in dollars) of 12 randomly chosen nursing supervisors are shown in the table at the left. At α=0.10, is there enough evidence to reject the claim that the standard deviation of the annual salaries is \$18,630?


tab2

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Step 1: Identify the claim and state the null hypothesis (H0) and the alternative hypothesis (Ha). The claim is that the standard deviation of the annual salaries is \(18,630. Thus, H0: σ = 18,630 (the standard deviation equals \)18,630), and Ha: σ ≠ 18,630 (the standard deviation does not equal \$18,630).
Step 2: Determine the critical value(s) and rejection region(s). Since the test involves the standard deviation and the population is normally distributed, use the chi-square distribution. The degrees of freedom (df) are calculated as n - 1, where n is the sample size. Here, n = 12, so df = 11. Using α = 0.10 for a two-tailed test, find the critical chi-square values from a chi-square table or calculator.
Step 3: Calculate the standardized test statistic X². First, compute the sample variance (s²) using the formula s² = Σ(x - x̄)² / (n - 1), where x̄ is the sample mean. Then, use the formula X² = (n - 1) * s² / σ² to calculate the test statistic, where σ² is the square of the claimed standard deviation.
Step 4: Compare the test statistic X² to the critical values. If X² falls within the rejection region (either below the lower critical value or above the upper critical value), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
Step 5: Interpret the decision in the context of the original claim. If the null hypothesis is rejected, conclude that there is enough evidence to reject the claim that the standard deviation of the annual salaries is \$18,630. If the null hypothesis is not rejected, conclude that there is not enough evidence to reject the claim.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (Ha), which represents the claim being tested. The goal is to determine whether there is enough evidence in the sample data to reject H0 in favor of Ha.
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Step 1: Write Hypotheses

Critical Value and Rejection Region

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the significance level (α), which indicates the probability of making a Type I error. The rejection region is the range of values for the test statistic that leads to the rejection of H0. If the calculated test statistic falls within this region, we reject the null hypothesis.
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Standard Deviation and Chi-Square Test

The standard deviation is a measure of the amount of variation or dispersion in a set of values. In the context of hypothesis testing for standard deviation, the Chi-Square test is used to determine if the sample standard deviation significantly differs from a hypothesized population standard deviation. The test statistic is calculated using the sample data, and its value is compared to the critical value to make a decision regarding the null hypothesis.
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Related Practice
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.


Repeat Customers A used textbook selling website claims that at least 60% of its new customers will return to buy their next textbook.

Textbook Question

Identifying the Nature of a Hypothesis Test In Exercises 37–42, state and in words and in symbols. Then determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed. Explain your reasoning. Sketch a normal sampling distribution and shade the area for the P-value.


Survey A polling organization reports that the number of responses to a survey mailed to 100,000 U.S. residents is not 100,000.

Textbook Question

Finding a P-Value In Exercises 13–18, find the P-value for the hypothesis test with the standardized test statistic z. Decide whether to reject H0 for the level of significance alpha.

Left-tailed test


z= 1.95

alpha=0.08

Textbook Question

Hypothesis Testing Using Rejection Regions In Exercises 7–12, (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 z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.


Changing Jobs A researcher claims that 40% of U.S. adults would consider changing jobs. In a random sample of 50 U.S. adults, 25 say they would consider changing jobs. At α=0.10, is there enough evidence to reject the researcher’s claim?

Textbook Question

Stating Hypotheses In Exercises 11–16, the statement represents a claim. Write its complement and state which is H0 and which is Ha.


μ < 128

Textbook Question

Stating Hypotheses In Exercises 11–16, the statement represents a claim. Write its complement and state which is H0 and which is Ha.


p = 0.21