Consider a situation involving the testing of processors in a production batch. Each processor is deemed acceptable if its performance meets certain specifications. The true proportion of processors meeting these criteria, denoted as , is currently unknown. A hardware engineer asserts that at least 80% of the processors in the batch adhere to the specified performance standards. To validate this claim, a random sample of 20 processors is chosen for testing. Let denote the number of acceptable processors among the 20 sampled. The engineer plans to dispute the claim that 0.8 if is less than or equal to 15. a. What is the PMF of for a given value of ? b. What is the probability that the claim is rejected when the actual value of is 0.8? Note that this is one type of error, the claim will be rejected when it is true. c. What is the probability of not rejecting the claim when = 0.7? Note that this is the other type of error, the claim ( 0.8) is not rejected even though it is not true. d. How do the "error probabilities" of (b) and (c) change if the value 15 in the decision rule is replaced by 14?