91Ó°ÊÓ

Let \(\mu\) be the true average weight of the ground beef in the packages. Then the null and alternative hypotheses for this test are: - Null Hypothesis \((H_0)\): \(\mu = 1\) - Alternative Hypothesis \((H_a)\): \(\mu \neq 1\) For part b: Perform the hypothesis test and find the p-value.

The test statistic we will use for this hypothesis test is the \(t\) -statistic since we have a small sample size (35) and the population standard deviation is unknown. The formula for calculating the \(t\) -statistic is: $$ t = \frac{\overline{x}-\mu_0}{(s/\sqrt{n})} $$ where \(\overline{x}\) is the sample mean, \(\mu_0\) is the hypothesized mean, \(s\) is the sample standard deviation, and \(n\) is the sample size. In this case, \(\overline{x} = 1.01\), \(\mu_0 = 1\), \(s = 0.18\), and \(n = 35\). Plugging these values into the formula, we get $$ t = \frac{1.01 - 1}{(0.18/\sqrt{35})} $$ Calculating the t-statistic: $$ t \approx 0.38 $$

Since the alternative hypothesis \((H_a)\) is a two-tailed test (\(\mu \neq 1\)), we need to find the probability of getting a \(t\) -statistic as extreme as the one calculated in the previous step. We use a two-tailed test, so the p-value is simply the sum of the two-tailed probabilities. Using a t-distribution table or software, we can find the p-value associated with the calculated t-statistic (\(0.38\)) and degrees of freedom \((n-1) = 34\). The p-value is approximately 0.71. For part c: As the quality control manager, explain the results of this hypothesis test to a consumer interest group.

We conducted a hypothesis test to check if the average amount of ground beef in the packages is indeed 1 pound. Our null hypothesis was that the average weight is equal to 1 pound, and our alternative hypothesis was that the average weight is not equal to 1 pound. The p-value we obtained from the hypothesis test is approximately 0.71. In order to make a decision about the null hypothesis (whether to reject it or fail to reject it), we need to compare the p-value to a predetermined significance level (usually denoted as \(\alpha\), with a common value of \(0.05\)). Since the p-value (\(0.71\)) is greater than the significance level (\(0.05\)), we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that the average amount of ground beef in the packages is different from 1 pound. In other words, we cannot say with enough confidence that the average weight of the packages is not 1 pound. To report this result to a consumer interest group, you could say: "Based on our sample of 35 packages of ground beef and a thorough hypothesis test, there is not enough evidence to suggest that the average weight of the packages is different from 1 pound. We will continue to monitor the weight of our packages to ensure quality and consistency."