5.5 #45 #47

37. Acceptance sampling is a method that uses small random samples from incoming shipments of products to assess the quality of the entire shipment. Typically, a random sample of size n is selected from a shipment, and each sampled item is tested to see whether it meets quality specifications. The number of sampled items that do not meet specifications is denoted by x. As long as x does not exceed a prespecified integer c, called the acceptance number, then the entire shipment is accepted for use. If x exceeds c, then the shipment is returned to the vendor. In practice, because n is usually small in comparison to the number of items in a shipment, a binomial distribution is used to describe the random variable x.

a. Suppose a company uses samples of size n 5 10 and an acceptance number of c 5 1 to evaluate shipments. If 10% of the items in a certain shipment are defective, what is the probability that this shipment will be returned to the vendor?

b. Suppose that a certain shipment contains no defective items. What is the probability that the shipment will be accepted by the sampling plan in part (a)?

c. Rework part (a) for shipments that are 5%, 20%, and 50% defective.

d. Let denote the proportion of defective items in a given shipment. Use your answers to parts (a)2(c) to plot the probability of accepting a shipment (on the vertical axis) against 5 0, .05, .10, .20, and .50 (on the horizontal axis). Connect the points on the graph with a smooth curve. The resulting curve is called the operating characteristic (OC) curve of the sampling plan. It gives a visual summary of how the plan performs for shipments of differing quality..

45. Refer to Exercise 37. Suppose that a large lot of items is inspected by taking a random sample of size n and determining the number x of defective items in the sample. The result is then reported in terms of the proportion p 5 xyn of defective items in the sample. Assume that the binomial distribution can be used to describe the behavior of the random variable x.

a. Suppose that 5% of the items in a particular lot are defective and that a random sample of size n 5 5 is to be taken from the lot. Calculate the probability that the sample proportion p falls within 1% of the true percent defective in the lot. That is, find p(.05 2 .01 # p # .05 1 .01)

b. Answer the question in part (a) for samples of size n 5 25.

c. Answer the question in part (a) for samples of size n 5 100. Hint: Use the normal approximation to the binomial distribution.

47. The Food and Drug Administration (FDA) oversees the approval of both medical devices and new drugs. To gain FDA approval, a new device must be shown to perform at least as well, and hopefully better, than any similar device already on the market. Suppose a medical device company develops a new system for connecting intravenous tubes used on hospital patients. To be comparable to an already-existing product, the force required to disconnect two tubes joined by the new device must not exceed 5 lb. To estimate the maximum force required to disconnect two tubes, several tests are made. For a random sample of n connections, the forces x 1 , x 2 , x 3 , . . . , x n required to disconnect the tubes are recorded and the maximum, M, of the n readings is used to estimate the maximum necessary force for all such connections.

a. Suppose that the actual distribution of forces needed to disconnect tubes can be described by a uniform distribution on the interval [2, 4]. For a sample of size n 5 2, do you expect the mean of the sampling distribution of M to be closer to 2 or 4?

b. Which do you expect to be larger, the mean of the sampling distribution of M for samples of size n52 or the mean of the sampling distribution of M for samples of size n5100? Use the definition of the maximum of a sample to justify your answer.

c. Will the variance of the sampling distribution of M for samples of size n52 be the same or different from the variance of the sampling distribution of M for samples of size n5100? Give a simple justification for your answer based on the definition of the sample maximum for n52 versus n5100.