A quality control inspector is always on the lookout for sub

A quality control inspector is always on the lookout for substandard parts and components provided to her manufacturing company by outside suppliers. Because most shipments contain some defective items, each must be subjected to inspection. Naturally, some shipments contain more defectives than others, and it is the job of the inspector to identify the most defective-laden shipments so that they may be returned to the supplier. Suppose the inspector selects a sample of n = 72 items from a given shipment for testing. Unbeknownst to the inspector, this particular shipment includes 9% defective components. If the policy is to return any shipment with at least 5% defectives, what is the probability that this particular shipment (with 9% defectives) will be accepted as good anyway?

Solution

n = 72

p = Prob for any random to be defective = 0.09

Returned if defects are >=0.05

X no of defectives is binomial with two outcomes

Suppose for 72 items if no of defectives isless than 3.6 then the shipment is accepted

As sample size is large let us take np as mean = 72(0.05) = 3.6

and variance =npq = 3.42

Std dev = 1.849

Sample mean = 0.09 *72 = 6.48

Std error = 1.849/rt 72 = 0.2179

Mean diff = 2.88

Hence P(accepting) = P(Z<2.88/0.2179) = P(Z<13.241) = 1.00