35. A test is made of $H_{0} ; \mu=50$ versus $H_{1}: \mu>50$. A sample of size $n=75$ is drawn, and $x=56$. The population standard deviation is $\sigma=20$. a. Compute the value of the test statistic $z$. b. Is $H_{0}$ rejected at the $\alpha=0.05$ level? c. Is $H_{0}$ rejected at the $\alpha=0.01$ level?

Step 1 ：Given that the sample mean \(x = 56\), the population mean \(\mu = 50\), the population standard deviation \(\sigma = 20\), and the sample size \(n = 75\).

Step 2 ：The formula for the z-score is given by \(z = \frac{(x - \mu)}{(\sigma / \sqrt{n})}\).

Step 3 ：Substitute the given values into the formula to calculate the z-score, which is approximately 2.60.

Step 4 ：The null hypothesis \(H_{0}\) is rejected if the z-score is greater than the critical value corresponding to the significance level \(\alpha\).

Step 5 ：The critical value for \(\alpha = 0.05\) is approximately 1.645 for a one-tailed test, and for \(\alpha = 0.01\) it is approximately 2.33.

Step 6 ：Compare the calculated z-score with these critical values to determine whether \(H_{0}\) is rejected.

Step 7 ：The calculated z-score is greater than the critical value of 1.645 for a significance level of \(\alpha = 0.05\), so the null hypothesis \(H_{0}\) would be rejected at this level.

Step 8 ：The z-score is less than the critical value of 2.33 for a significance level of \(\alpha = 0.01\), so the null hypothesis would not be rejected at this level.

Step 9 ：Final Answer: a. The value of the test statistic \(z\) is approximately \(\boxed{2.60}\). b. Yes, \(H_{0}\) is rejected at the \(\alpha=0.05\) level. c. No, \(H_{0}\) is not rejected at the \(\alpha=0.01\) level.