A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 47 tablets, then accept the whole batch if there is only one or none that doesn't meet the required specifications. If one shipment of 3000 aspirin tablets actually has a 3% rate of defects, what is the probability that this whole shipment will be accepted? Will almost all such shipments be accepted, or will many be rejected?

Question

07-03-2020
Mathematics

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A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 47 tablets, then accept the whole batch if there is only one or none that doesn't meet the required specifications. If one shipment of 3000 aspirin tablets actually has a 3% rate of defects, what is the probability that this whole shipment will be accepted? Will almost all such shipments be accepted, or will many be rejected?

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warylucknow warylucknow · Answer 1 · 2020-03-11T06:38:17+01:00

Answer:

The probability that the whole shipment will be accepted is 0.5862.

Step-by-step explanation:

Let X = number of the aspirin tablets that doesn't meet the required specifications.

The probability of the random variable X is, P (X) = p = 0.03.

The sample of n = 47 tablets are tested from each batch.

The probability of any of the tablets being defective is independent of the others.

The probability mass function of X is,

[tex]P(X=x)={n\choose x}p^{x}(1-)^{n-x};\ x=0,1,2,3...[/tex]

Compute the probability that the whole shipment of 3000 tablets will be accepted as follows:

P (X ≤ 1) = P (X = 0) + P (X = 1)

[tex]={47\choose 0}0.03^{0}(1-0.03)^{47-0}+{47\choose 1}0.03^{1}(1-0.03)^{47-1}\\=0.2389+0.3473\\=0.5862[/tex]

Thus, the probability that the whole shipment will be accepted is 0.5862.

The sample of 47 tablets is significantly small when drawn from a population of 3000 tablets. So it is difficult to make conclusion about all such shipments of aspirin tablets.