A plan for an executive travelers’ club has been developed by an airline on the premise that 10% of its current customers would qualify for membership. a.? ?Assuming the validity of this premise, among 25 randomly selected current customers, what is the probability that between 2 and 6 (inclusive) qualify for membership? b.? ?Again assuming the validity of the premise, what are the expected number of customers who qualify and the standard deviation of the number who qualify in a random sample of 100 current customers? c.? ?Let ?X ?denote the number in a random sample of 25 current customers who qualify for membership. Consider rejecting the company’s premise in favor of the claim that p > .10 if x ? 7. What is the probability that the company’s premise is rejected when it is actually valid? d.? ?Refer to the decision rule introduced in part (c). What is the probability that the company’s premise is not rejected even though p = .20 (i.e., 20% qualify)?

Problem 106E Answer: Step1: We have A plan for an executive travelers’ club has been developed by an airline on the premise that 10% of its current customers would qualify for membership. We need to find, a. Assuming the validity of this premise, among 25 randomly selected current customers, what is the probability that between 2 and 6 (inclusive) qualify for membership b. Again assuming the validity of the premise, what are the expected number of customers who qualify and the standard deviation of the number who qualify in a random sample of 100 current customers c. Let X denote the number in a random sample of 25 current customers who qualify for membership. Consider rejecting the company’s premise in favor of the claim that p > 0.10 if x 7. What is the probability that the company’s premise is rejected when it is actually valid d. Refer to the decision rule introduced in part (c). What is the probability that the company’s premise is not rejected even though p = .20 (i.e., 20% qualify) Step2: a). We have n = 25 and p = 10% = 0.10 Let “X” be random variable it follows binomial distribution with parameter “n, p” X ~B(n, p) X ~B(25, 0.1) The probability mass function of X is given by n x n-x p(x) = C x q , x = 0,1,2,...,n With mean E(x) = np and Var(x) = np(1 - p) The probability that between 2 and 6 qualify for membership is given by P(between 2 and 6 qualify for membership) = P(2X6) P(2X6) = P(X = 2) + P(X = 3) + ….+ P(X = 6) = { C (0.1) (0.9)252+ C (0.1) (0.9) 253 + C (0.1) (0.9) 254+….+ C5 2 6 256 3 4 6 (0.1) (0.9) } P(2X6) i s obtained from Excel by using the formula “=Binomdist(x,n,p,FALSE)” “=Binomdist(0 to 6,25,0.1,FALSE)” x P(2X6) 2 0.265888144 3 0.226497308 4 0.138415021 5 0.064593677 6 0.023923584 sum 0.719317733 Therefore,The probability that between 2 and 6 qualify for membership is 0.7193. Step3: b). We have n = 100 and p = 0.1, q = 0.9 The expected number of customers who qualify in a random sample of 100 current customers is given by E(X) = np = 100×0.1 = 10 Therefore, The expected number of customers who qualify in a random sample of 100 current customers is 10. var(X) = npq = 100×0.1×0.9 = 9 The standard deviation of the number who qualify in a random sample of 100 current customers is given by x nq = 100 × 0.1 × 0.9 = 3. Therefore, The standard deviation of the number who qualify in a random sample of 100 current customers is 3. Step4: c). Let X denote the number in a random sample of 25 current customers who qualify for membership.Consider rejecting the company’s premise in favor of the claim that p > 0.10 if x 7. We have n = 25 and p = 0.10 , q = 0.9 Now, The probability that the company’s premise is rejected when it is actually valid is given by P(X 7) = 1 - P(X 6) P(X 6) i s obtained from Excel by using the formula “=Binomdist(x,n,p,FALSE)” “=Binomdist(0 to 6,25,0.1,FALSE)” x P(X 6) 0 0.071789799 1 0.199416108 2 0.265888144 3 0.226497308 4 0.138415021 5 0.064593677 6 0.023923584 Sum P(X 6) = 0.990523639 Hence, P(X 7) = 1 - P(X 6) = 1 - 0.9905 = 0.0095. Therefore, The probability that the company’s premise is rejected when it is actually valid is 0.0095. Step5: d). We have = 25 and p = 0.20 , q = 0.8 The probability that the company’s premise is not rejected even though it has 20% quality is given by P(the company’s premise is not rejected even though it has 20% quality) = P(X 6) P(X 6) i s obtained from Excel by using the formula “=Binomdist(x,n,p,FALSE)” “=Binomdist(0 to 6,25,0.2,FALSE)” x P(X 6) 0 0.003777893 1 0.023611832 2 0.070835497 3 0.135768036 4 0.18668105 5 0.196015103 6 0.163345919 Sum P(X 6) = 0.780035331 Therefore, The probability that the company’s premise is not rejected even though it has 20% quality is 0.7800.