Data on the weights (lb) 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 0.05 significance level for both parts.
\begin{tabular}{|c|c|c|}
\hline & Diet & Regular \\
\hline $\boldsymbol{\mu}$ & $\boldsymbol{\mu}_{1}$ & $\boldsymbol{\mu}_{2}$ \\
\hline $\mathbf{n}$ & 38 & 38 \\
\hline$\overline{\mathbf{x}}$ & $0.79613 \mathrm{lb}$ & $0.81202 \mathrm{lb}$ \\
\hline $\mathbf{s}$ & $0.00444 \mathrm{lb}$ & $0.00741 \mathrm{lb}$ \\
\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.
\[
\begin{array}{l}
H_{0}: \mu_{1} \neq \mu_{2} \\
H_{1}: \mu_{1}< \mu_{2}
\end{array}
\]
C.
\[
\begin{array}{l}
H_{0}: \mu_{1}=\mu_{2} \\
H_{1}: \mu_{1}> \mu_{2}
\end{array}
\]
B.
\[
\begin{array}{l}
H_{0}: \mu_{1}=\mu_{2} \\
H_{1}: \mu_{1} \neq \mu_{2}
\end{array}
\]
D.
\[
\begin{array}{l}
H_{0}: \mu_{1}=\mu_{2} \\
H_{1}: \mu_{1}< \mu_{2}
\end{array}
\]
The test statistic, $\mathrm{t}$, is
(Round to two decimal places as needed.)
Answer
So, the final answer is: The null and alternative hypotheses are \[\begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1}<\mu_{2} \end{array}\] and the test statistic, t, is \(\boxed{-11.34}\).
Step 1 ：The problem is asking to 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. This is a hypothesis testing problem for two independent samples.
Step 2 ：The null hypothesis is that the means of the two populations are equal, and the alternative hypothesis is that the mean of the diet soda is less than the mean of the regular soda. So, the null and alternative hypotheses are \[\begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1}<\mu_{2} \end{array}\]
Step 3 ：We can use the formula for the t-test statistic for two independent samples to calculate the test statistic. The given values are: \[n1 = 38, \quad x_{\bar{1}} = 0.79613, \quad s1 = 0.00444, \quad n2 = 38, \quad x_{\bar{2}} = 0.81202, \quad s2 = 0.00741\]
Step 4 ：The test statistic t is approximately -11.34. This value is negative, which indicates that the mean weight of the diet soda is less than the mean weight of the regular soda, as expected.
Step 5 ：So, the final answer is: The null and alternative hypotheses are \[\begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1}<\mu_{2} \end{array}\] and the test statistic, t, is \(\boxed{-11.34}\).
