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Ch. 10 - Chi-Square Tests and the F-Distribution
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. 10 - Chi-Square Tests and the F-DistributionProblem 10.R.17
Chapter 10, Problem 10.R.17

"In Exercises 17–20, (a) identify the claim and state H₀ and Hₐ, (b) find the critical value and identify the rejection region, (c) find the test statistic F, (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 samples are random and independent, and the populations are normally distributed.


A travel consultant claims that the standard deviations of hotel room rates for Sacramento, CA, and San Francisco, CA, are the same. A sample of 36 hotel room rates in Sacramento has a standard deviation of \$51 and a sample of 31 hotel room rates in San Francisco has a standard deviation of \$37. At α=0.10, can you reject the travel consultant’s claim? (Adapted from Expedia)"

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1
Identify the claim and state the null hypothesis (H₀) and the alternative hypothesis (Hₐ): The claim is that the standard deviations of hotel room rates for Sacramento and San Francisco are the same. This translates to H₀: σ₁² = σ₂² (the variances are equal) and Hₐ: σ₁² ≠ σ₂² (the variances are not equal). This is a two-tailed test.
Find the critical value and identify the rejection region: Use the F-distribution table with degrees of freedom df₁ = n₁ - 1 = 36 - 1 = 35 and df₂ = n₂ - 1 = 31 - 1 = 30. Since α = 0.10 and this is a two-tailed test, divide α by 2 for each tail (α/2 = 0.05). Look up the critical F-values for both the upper and lower tails. The rejection region will be F < F_lower or F > F_upper.
Calculate the test statistic F: The formula for the F-test statistic is F = (s₁² / s₂²), where s₁ and s₂ are the sample standard deviations. Substitute s₁ = 51 and s₂ = 37 into the formula. Compute F = (51² / 37²).
Decide whether to reject or fail to reject the null hypothesis: Compare the calculated F value to the critical values from step 2. If F falls into the rejection region (either F < F_lower or F > F_upper), reject H₀. Otherwise, fail to reject H₀.
Interpret the decision in the context of the original claim: If you rejected H₀, conclude that there is sufficient evidence to reject the travel consultant’s claim that the standard deviations are the same. If you failed to reject H₀, 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 indicates the presence of an effect or difference. In this case, the null hypothesis would state that the standard deviations of hotel room rates in Sacramento and San Francisco are equal.
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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. For this problem, the critical value will be determined using the F-distribution, as the test involves comparing variances.
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F-Test for Equality of Variances

The F-test is a statistical test used to compare the variances of two populations to determine if they are significantly different. It calculates the F-statistic by taking the ratio of the two sample variances. In this scenario, the F-statistic will be computed using the sample standard deviations from Sacramento and San Francisco, and compared against the critical value to decide whether to reject the null hypothesis regarding the equality of variances.
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Related Practice
Textbook Question

In Exercises 17–20, (a) identify the claim and state H₀ and Hₐ, (b) find the critical value and identify the rejection region, (c) find the test statistic F, (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 samples are random and independent, and the populations are normally distributed.


[APPLET] An instructor claims that the variance of SAT evidence-based reading and writing scores is different than the variance of SAT math scores. The table shows the SAT evidence-based reading and writing scores for 12 randomly selected students and the SAT math scores for 12 randomly selected students. At α=0.01, can you support the instructor’s claim?


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Textbook Question

In Exercises 21 and 22, (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal.

[APPLET] The table shows the monthly electric bills (in dollars) for a sample of households from four regions of the United States. At α=0.10, can you conclude that the mean monthly electric bill is different in at least one of the regions? (Adapted from U.S. Energy Information Administration)

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Textbook Question

In Exercises 21 and 22, (c) find the test statistic F, Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal.

[APPLET] The table shows the monthly electric bills (in dollars) for a sample of households from four regions of the United States. At α=0.10, can you conclude that the mean monthly electric bill is different in at least one of the regions? (Adapted from U.S. Energy Information Administration)

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Textbook Question

In Exercises 13–16, find the critical F-value for a two-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.


α=0.01,d.f.N=11,d.f.D=13

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Textbook Question

"In Exercises 9–12, find the critical F-value for a right-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.


α=0.01,d.f.N=12,d.f.D=10"

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Textbook Question

In Exercises 5–8, (a) find the expected frequency for each cell in the contingency table, (b) identify the claim and state H0 and Ha, (c) determine the degrees of freedom, find the critical value, and identify the rejection region, (d) find the chi-square test statistic, (e) decide whether to reject or fail to reject the null hypothesis, and (f) interpret the decision in the context of the original claim.


The contingency table shows the distribution of a random sample of fatal pedestrian and bicyclist motor vehicle collisions by time of day in a recent year. At α=0.10, can you conclude that the type of crash victim and the time of day are related? (Adapted from National Highway Traffic Safety Administration)


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