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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.21e
Chapter 10, Problem 10.R.21e

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)
Table showing monthly electric bills in dollars for households from four U.S. regions: Northeast, Midwest, South, and West.

Verified step by step guidance
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Step 1: Identify the problem type and hypotheses. This is a one-way ANOVA test because we are comparing the means of more than two independent groups (Northeast, Midwest, South, West). The null hypothesis \(H_0\) is that all group means are equal: \(\mu_{Northeast} = \mu_{Midwest} = \mu_{South} = \mu_{West}\). The alternative hypothesis \(H_a\) is that at least one group mean is different.
Step 2: 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 \(i\) (Northeast, Midwest, South, West).
Step 3: Compute the overall mean \(\bar{x}\) by combining all data points from the four groups. Then calculate the Between-Group Sum of Squares (SSB) and Within-Group Sum of Squares (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 4: Calculate the Mean Squares: Mean Square Between (MSB) and Mean Square Within (MSW) by dividing the sums of squares by their respective degrees of freedom: \(MSB = \frac{SSB}{k-1}\) \(MSW = \frac{SSW}{N-k}\) where \(N\) is the total number of observations.
Step 5: Compute the F-statistic as \(F = \frac{MSB}{MSW}\). Compare this F-value to the critical value from the F-distribution table at significance level \(\alpha = 0.10\) with degrees of freedom \(df_1 = k-1\) and \(df_2 = N-k\). If \(F\) is greater than the critical value, reject the null hypothesis and conclude that there is sufficient evidence to say the mean monthly electric bill differs in at least one region. Otherwise, fail to reject the null hypothesis.

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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 from the others. It tests the null hypothesis that all group means are equal against the alternative that at least one differs. This method is appropriate here because we are comparing electric bills across four regions.
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Introduction to ANOVA

Assumptions of ANOVA

ANOVA relies on key assumptions: samples must be random and independent, populations should be normally distributed, and population variances must be equal (homogeneity of variance). These assumptions ensure the validity of the test results and affect the interpretation of the p-value and conclusions.
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ANOVA Test

Significance Level (α) and Hypothesis Testing

The significance level α (here 0.10) is the threshold for deciding whether to reject the null hypothesis. If the p-value from ANOVA is less than α, we conclude there is sufficient evidence that at least one region's mean electric bill differs. This decision must be interpreted in the context of the original claim about differences among regions.
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Performing Hypothesis Tests: Proportions
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, (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, the populations are normally distributed, and the population variances are equal.


[APPLET] The table shows the annual incomes (in dollars) for a sample of families from four regions of the United States. At α=0.05, can you conclude that the mean annual income of families is different in at least one of the regions? (Adapted from U.S. Census Bureau)


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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=0.10,d.f.N=15,d.f.D=27"

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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)

39
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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 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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