In Exercises 21 and 22, (d) decide whether to reject or fail to reject the null hypothesis,
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

1. Step 1: State the hypotheses. The null hypothesis \(H_0\) is that the mean monthly electric bills are equal across all four regions: \(\mu_{Northeast} = \mu_{Midwest} = \mu_{South} = \mu_{West}\). The alternative hypothesis \(H_a\) is that at least one region's mean monthly electric bill is different.
2. Step 2: Since the problem involves comparing means from more than two groups, use a one-way ANOVA test. Calculate the group means and the overall mean from the data provided for each region.
3. Step 3: Calculate the Sum of Squares Between Groups (SSB) and the Sum of Squares Within Groups (SSW). SSB measures the variation between the group means and the overall mean, while SSW measures the variation within each group.
4. Step 4: Compute the Mean Square Between Groups (MSB) and Mean Square Within Groups (MSW) by dividing SSB and SSW by their respective degrees of freedom. Then calculate the F-statistic as \(F = \frac{MSB}{MSW}\).
5. Step 5: Compare the calculated F-statistic to the critical value from the F-distribution table at \(\alpha = 0.10\) with appropriate degrees of freedom. If \(F\) is greater than the critical value, reject the null hypothesis; otherwise, fail to reject it.

Key Concepts

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 against the alternative that at least one differs. ANOVA uses the F-distribution to assess variance between groups relative to variance within groups.

Null and Alternative Hypotheses

The null hypothesis (H0) assumes no difference in the population means across the groups, while the alternative hypothesis (Ha) suggests that at least one group mean is different. Deciding to reject or fail to reject H0 depends on the p-value or test statistic compared to the significance level (α). This decision guides conclusions about the data.

Assumptions of ANOVA

ANOVA requires certain 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, so checking these conditions is essential before performing ANOVA.