(a) (d) (a) You want to test the hypothesis that the sample mean, X, is 50. Qther- wise, it is greater than 50. Your test can be written as the following: Ho: = 50 HA: > 50 and a = 0.05, the random sample is of size 10 taken from a population that is N(, 250). Indicate what is the distribution of X. What is the critical value c? What is the probability of rejecting the null when the null is true? What is the probability of failing to reject the null when the null is true? 2. A researcher reports survey results by stating that the standard error of the mean is 20. The population standard deviation is 500. What is the probability of rejecting the null when = 60, i.e., when the null false? How large was the sample used in this survey? What is the probability that the point estimate was within 25 of the population mean?