Problem
Data on the weights $(\mathrm{Ib})$ of the contents of cans of diet soda versus the contents of cans of the regular version of the soda is summarized to the right. 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) below. Use a 0.01 significance level for both parts. \begin{tabular}{|c|c|c|} \hline & Diet & Regular \\ \hline $\boldsymbol{\mu}$ & $\mu_{1}$ & $\mu_{2}$ \\ \hline $\mathbf{n}$ & 24 & 24 \\ \hline$\overline{\mathbf{x}}$ & $0.78265 \mathrm{lb}$ & $0.81268 \mathrm{lb}$ \\ \hline $\mathbf{s}$ & $0.00437 \mathrm{lb}$ & $0.00742 \mathrm{lb}$ \\ \hline \end{tabular} a. Test the claim that the contents of cans of diet soda have weights with a mean that is less than the mean for the regular soda. What are the null and alternative hypotheses? A. \[ \begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1}>\mu_{2} \end{array} \] c. \[ \begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1} \neq \mu_{2} \end{array} \] B. \[ \begin{array}{l} H_{0}: \mu_{1} \neq \mu_{2} \\ H_{1}: \mu_{1}<\mu_{2} \end{array} \] D. \[ \begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1}<\mu_{2} \end{array} \]
Solution
Step 1 ：The null and alternative hypotheses are given by: \[H_{0}: \mu_{1}=\mu_{2}\] \[H_{1}: \mu_{1}<\mu_{2}\]
Step 2 ：The final answer is: \[\boxed{H_{0}: \mu_{1}=\mu_{2}, H_{1}: \mu_{1}<\mu_{2}}\]
