A start-up cell phone applications company is interested in determining whether household incomes are different for subscribers to two different service providers. A random sample of 27 subscribers to each of the two service providers was taken, and the annual household income for each subscriber was recorded. The partially completed ANOVA table for the analysis is below. Complete parts a and b below. Click the icon to view the partially completed ANOVA table. b. Based on the sample results, can the start-up firm conclude that there is a difference in household incomes for subscribers to the two service providers? You may assume normal distributions and equal variances. Conduct your test at the $\alpha=0.10$ level of significance. Be sure to state a critical F-statistic, a decision rule, and a conclusion. Identify the hypotheses for the test. Choose the correct answer below. A. $\mathrm{H}_{0}: \mu_{1} \neq \mu_{2}$ $\mathrm{H}_{\mathrm{A}}$ : All of the population means are equal C. $H_{0}: \mu_{1} \neq \mu_{2}$ $\mathrm{H}_{\mathrm{A}}$ : At least two of the population means are equal B. $H_{0}: \mu_{1}=\mu_{2}$ $\mathrm{H}_{\mathrm{A}}$ : At least two of the population means are different D. $H_{0}: \mu_{1}=\mu_{2}$ $\mathrm{H}_{\mathrm{A}}:$ All of the population means are different State the decision rule at the $\alpha=0.10$ level of significance. Select the correct choice below and fill in any answer boxes in your choice. (Round to four decimal places as needed.) A. If $\mathrm{F}<\square$ or $\mathrm{F}>\square$, reject the null hypothesis. Otherwise, do not reject. B. If $\mathrm{F}<\square$, reject the null hypothesis. Otherwise, do not reject. C. If $F>\square$, reject the null hypothesis. Otherwise, do not reject.
Solution
Step 1 ：The correct hypotheses for the test are: B. $H_{0}: \mu_{1}=\mu_{2}$ $\mathrm{H}_{\mathrm{A}}$ : At least two of the population means are different The null hypothesis (H0) is that there is no difference in the household incomes for subscribers to the two different service providers, i.e., the population means are equal. The alternative hypothesis (HA) is that there is a difference in the household incomes for subscribers to the two different service providers, i.e., at least two of the population means are different. The decision rule at the $\alpha=0.10$ level of significance is: C. If $F>\square$, reject the null hypothesis. Otherwise, do not reject. The F-statistic is used in the analysis of variance (ANOVA). If the calculated F-statistic is greater than the critical F-value (which can be found in the F-distribution table based on the degrees of freedom and the level of significance), we reject the null hypothesis. The exact critical F-value would depend on the degrees of freedom, which are not provided in the question. Without the actual data or the partially completed ANOVA table, we cannot calculate the F-statistic or make a conclusion about the null hypothesis. However, the decision rule and hypotheses stated above are correct based on the information provided in the question.
