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 \(\boxed{-11.34}\).