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

Larson 8th Edition

Ch. 10 - Chi-Square Tests and the F-Distribution

Problem 10.R.21c

Chapter 10, Problem 10.R.21c

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)

Verified step by step guidance

Step 1: Identify the problem type. This is a one-way ANOVA test because we are comparing the means of monthly electric bills across four different regions (Northeast, Midwest, South, West).

Step 2: State the hypotheses. The null hypothesis $H_0$ is that all region means are equal: $\mu_{Northeast} = \mu_{Midwest} = \mu_{South} = \mu_{West}$. The alternative hypothesis $H_a$ is that at least one region mean is different.

Step 3: Calculate the sample means and variances for each region using the data provided. This involves computing the mean $\bar{x}_i$ and variance $s_i^2$ for each group $i$.

Step 4: Compute the overall mean of all data combined. Then calculate the Sum of Squares Between Groups (SSB) and Sum of Squares Within Groups (SSW) using the formulas: $SSB = \sum_{i=1}^k n_i (\bar{x}_i - \bar{x})^2$ $SSW = \sum_{i=1}^k (n_i - 1) s_i^2$ where $k$ is the number of groups and $n_i$ is the sample size of group $i$.

Step 5: Calculate the Mean Squares Between (MSB) and Mean Squares Within (MSW): $MSB = \frac{SSB}{k-1}$ $MSW = \frac{SSW}{N-k}$ where $N$ is the total number of observations. Then compute the test statistic $F$ as: $F = \frac{MSB}{MSW}$. This $F$ value will be compared to the critical value from the $F$ distribution with degrees of freedom $k-1$ and $N-k$ to decide whether to reject $H_0$.

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Key Concepts

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

Analysis of Variance (ANOVA)

ANOVA is a statistical method used to compare the means of three or more groups to determine if at least one group mean is significantly different. It tests the null hypothesis that all group means are equal by analyzing variance within and between groups. This method is appropriate when comparing multiple populations, such as electric bills across different regions.

F-Test Statistic

The F-test statistic in ANOVA measures the ratio of variance between group means to the variance within the groups. A larger F value suggests greater differences among group means relative to variability within groups. Calculating the F statistic helps decide whether to reject the null hypothesis at a given significance level (α).

Assumptions of ANOVA

ANOVA relies on key assumptions: samples must be random and independent, populations normally distributed, and population variances equal (homogeneity of variance). These assumptions ensure the validity of the F-test results. Violations can lead to incorrect conclusions about differences among group means.

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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 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 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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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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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=0.10,d.f.N=5,d.f.D=12"

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