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. Aslong as x does not exceed a pre specified 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 = 10 and an acceptance number of c = 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 t denote the proportion of defective items in a given shipment. Use your answers toparts (a)-(c) to plot the probability of accepting a shipment (on the vertical axis) against Tt=0, .05, .10, .20, and .50 (on the horizontal axis). Connect the points on the graph with a smoothcurve. 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.

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