3. An electronic component for a medical Xray unit is produced in lots of size N

Question

3. An electronic component for a medical Xray unit is produced in lots of size N = 25. An acceptance testing procedure is used by the purchaser to protect against lots that contain too many nonconforming components. The procedure consists of selecting five components at random from the lot (without replacement) and testing them. If none of the components is nonconforming, the lot is accepted. (20 points)

a. If the lot contains two nonconforming components, what is the probability of lot acceptance? (5 points)

b. Calculate the desired probability in (a) using the binomial approximation. Is this approximation satisfactory? Why or why not?  (5 points)

c. Suppose the lot size was N = 150. Would the binomial approximation be satisfactory in this case? (5 points)

d. Suppose that the purchaser will reject the lot with the decision rule of finding one or more nonconforming components in a sample of size n, and wants the lot to be rejected with probability at least 0.95 if the lot contains five or more nonconforming components. How large should the sample size n be? (5 points)

Explanation / Answer

Ans:

a) Assume that the lot actually contains two nonconforming components, and let Y = no. of nonconforming components in the sample. Then Y ~ Hypergeometric(N = 25, r = 2, n = 5).

What is the probability of lot acceptance?

The probability that no nonconforming units will be found in the sample is 23C5*2C0/25C5

=23*22*21*20*19/25*24*23*22*21=0.6333

b) p=2/25=0.08

n=5

np=0.08*5=2

For binomial approximation,np>=5,so not satisfactory or valid approximation.

P(k=0)=5C0 0.080*0.925=0.6591

c)p=2/150=0.013

n=5

np=5*0.013=0.065<5

so not valid approximation.

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