An experiment was conducted to determine whether giving candy to dining parties resulted in greater tips. The mean tip percentages and standard deviations are given in the accompanying table along with the sample sizes. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Complete parts (a) and (b). \begin{tabular}{|c|c|c|c|c|} \hline & $\boldsymbol{\mu}$ & $\mathbf{n}$ & $\overline{\mathbf{x}}$ & $\mathbf{s}$ \\ \hline No candy & $\boldsymbol{\mu}_{1}$ & 29 & 19.46 & 1.49 \\ \hline Two candies & $\boldsymbol{\mu}_{2}$ & 29 & 22.37 & 2.37 \\ \hline \end{tabular} a. Use a 0.05 significance level to test the claim that giving candy does result in greater tips. What are the null and alternative hypotheses? A. $H_{0}: \mu_{1}=\mu_{2}$ $H_{1}: \mu_{1}>\mu_{2}$ C. \[ \begin{array}{l} H_{0}: \mu_{1} \neq \mu_{2} \\ H_{1}: \mu_{1}<\mu_{2} \end{array} \] B. \[ \begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1}<\mu_{2} \end{array} \] D. \[ \begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1} \neq \mu_{2} \end{array} \]
Solution
Step 1 ：The question is asking us to test the claim that giving candy results in greater tips. This means we are comparing the mean tip percentage of the group that received no candy to the group that received two candies.
Step 2 ：The null hypothesis is typically a statement of no effect or no difference. In this case, it would be that the mean tip percentage is the same for both groups, or \(\mu_{1} = \mu_{2}\).
Step 3 ：The alternative hypothesis is what we are testing for. In this case, we are testing for the claim that giving candy results in greater tips. This would mean that the mean tip percentage for the group that received two candies is greater than the group that received no candy, or \(\mu_{1} < \mu_{2}\).
Step 4 ：So, the null and alternative hypotheses are: \(H_{0}: \mu_{1}=\mu_{2}\) and \(H_{1}: \mu_{1}<\mu_{2}\).
Step 5 ：This corresponds to option B in the given choices.
Step 6 ：Final Answer: The correct choice is \(\boxed{B}\).
