A magazine claims that the mean amount spent by a customer at Burger Stop is greater than the mean amount spent by a customer at Fry World. The results for samples of customer transactions for the two fast food restaurants are shown below. At $\alpha=0.05$, can you support the magazine's claim? Assume the population variances are equal. Assume the samples are random and independent, and the populations are normally distributed. Complete parts (a) through (e) below. \begin{tabular}{c|c} Burger Stop & Fry World \\ \hline $\bar{x}_{1}=\$ 9.35$ & $\bar{x}_{2}=\$ 8.72$ \\ $\mathrm{~s}_{1}=\$ 0.87$ & $\mathrm{~s}_{2}=\$ 0.84$ \\ $\mathrm{n}_{1}=13$ & $\mathrm{n}_{2}=11$ \end{tabular} The null hypothesis, $H_{0}$, is $\mu_{1} \leq \mu_{2}$. The altemative hypothesis, $H_{a}$, is $\mu_{1}>\mu_{2}$. Which hypothesis is the claim? The alternative hypothesis, $\mathrm{H}_{\mathrm{a}}$ The null hypothesis, $\mathrm{H}_{0}$ (b) Find the critical value(s) and identify the rejection region(s). Enter the critical value(s) below. (Type an integer or decimal rounded to three decimal places as needed. Use a comma to separate answers as needed.)
Solution
Step 1 ：The null hypothesis, \(H_{0}\), is \(\mu_{1} \leq \mu_{2}\). The alternative hypothesis, \(H_{a}\), is \(\mu_{1}>\mu_{2}\). The claim is the alternative hypothesis, \(H_{a}\).
Step 2 ：The test is a right-tailed test because the alternative hypothesis is \(\mu_{1}>\mu_{2}\).
Step 3 ：The critical value can be found using the standard normal distribution (Z-distribution) because we are assuming the population variances are equal.
Step 4 ：The critical value is the z-score such that the area to the right of it is equal to the significance level, \(\alpha=0.05\).
Step 5 ：The rejection region is the set of values greater than the critical value.
Step 6 ：The critical value for a right-tailed test at a significance level of 0.05 is approximately 1.645. This means that we will reject the null hypothesis if our test statistic is greater than 1.645.
Step 7 ：Final Answer: The critical value is \(\boxed{1.645}\). The rejection region is all values greater than \(\boxed{1.645}\).
