Dole Pineapple Inc. is concerned that the 16-ounce can of sliced pineapple is being overfilled. Assume the population standard deviation of the process is 0.03 ounce. The quality-control department took a random sample of 50 cans and found that the arithmetic mean weight was 16.05 ounces. At the $5 \%$ level of significance, can we conclude that the mean weight is greater than 16 ounces? The following $z$-values are provided. \begin{tabular}{|c|c|} \hline$P(x \leq z)$ & $z$-value \\ \hline 0.01 & -2.326 \\ \hline 0.05 & -1.645 \\ \hline 0.10 & -1.282 \\ \hline 0.90 & 1.282 \\ \hline 0.95 & 1.645 \\ \hline 0.99 & 2.326 \\ \hline \end{tabular} A. State the null and alternate hypotheses. B. Compute the test statistic. C. Compute the p-value. D. What is your decision regarding the null hypothesis?

Step 1 ：State the null and alternate hypotheses. The null hypothesis, denoted \(H_0\), is that the mean weight of the cans is equal to 16 ounces. The alternate hypothesis, denoted \(H_a\), is that the mean weight of the cans is greater than 16 ounces. In mathematical terms, we can express these as: \(H_0: \mu = 16\) \(H_a: \mu > 16\)

Step 2 ：Compute the test statistic. The test statistic is a z-score (z). The z-score is the difference between the sample mean and the population mean divided by the standard deviation of the population. The formula to calculate the z-score is: \(z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}\) where \(\bar{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the standard deviation of the population, and \(n\) is the size of the sample. Substituting the given values we get: \(z = \frac{16.05 - 16}{0.03/\sqrt{50}}\)

Step 3 ：Compute the p-value. The p-value is the probability that the observed data would occur if the null hypothesis were true. We can find the p-value using the z-score and the standard normal distribution table. The p-value is the area to the right of the z-score on the standard normal distribution curve. Since we are testing for a mean greater than 16 ounces, we want the area to the right of the z-score. The p-value is: \(P(Z > z)\)

Step 4 ：Make a decision regarding the null hypothesis. If the p-value is less than the level of significance (0.05), we reject the null hypothesis. If the p-value is greater than the level of significance, we fail to reject the null hypothesis. Based on our p-value, we make our decision.