tStandardized Normal Probahilitiee 14 0.or Problem 6: Proportion Defectives and

Question

tStandardized Normal Probahilitiee 14 0.or Problem 6: Proportion Defectives and Acceptance Sampling Apple Products Inc, an apple products packaging firm and its suppliers have agreed to an incoming inspection procedure called "Acceptance Sampling. Under this scheme, when a truckload of apples arrives at the firm's packaging plant, a random sample of 150 apples is selected and is inspected for bruises discoloration, and other defects. The percentage defective rate is calculated. The entire truckload of apples will be rejected if more than 5% of the sample is defective. Otherwise the entire truck load is accepted. Suppose a truck arrives with a population defective rate 8% (ie, 8% of the apples do not meet the acceptance standard). What is the probability that this truck will be accepted under the inspection scheme? (In Quality Control jargon, this is called the "Consumer's Risk"- Quality of the truckload is not acceptable, but under sampling there is this much chance that the (a) truckload is accepted.) Suppose another truck arrives with a population defective rate 4% (ie, 4% of the apples do not meet the acceptance standard). What is the probability that this truck will be rejected under the inspection scheme? (In Quality Control jargon, this is called the "Producer's Risk"- Quality of the truckload (b) is acceptable, but under sampling there is this much chance that the truckload is rejected)

Explanation / Answer

Here probability of population defective rate of new truck = 0.08

Here sample size = 150

Here the whole population of apples will be accepted when number of defective apples would be less than 0.05.

Stanadrd error of proportion sep = sqrt [0.08 * 0.92/150] = 0.02215

Here,

Pr(Apples population will be accepted) = Pr(p < 0.05) = NORM (p < 0.05 ; 0.08; 0.02215)

Z = (0.05 - 0.08)/0.02215 = -1.3544

Pr(p < 0.05) = NORM (p < 0.05 ; 0.08; 0.02215) = Pr(Z < -1.3544) = 0.0878

(b)  probability of population defective rate of new truck = 0.04

Here the whole population of apples will be rejected when number of defective apples would be greater than 0.05.

Stanadrd error of proportion sep = sqrt [0.04* 0.96/150] = 0.016

Pr(Apples population will be rejected) = Pr(p > 0.05) = 1 - NORM (p < 0.05 ; 0.08; 0.02215)

Z = (0.05 - 0.04)/0.016 = 0.625

Pr(p > 0.05) = 1- NORM (p < 0.05 ; 0.08; 0.02215) = 1 - Pr(Z < 0.625) = 1 - 0.7502 = 0.2498

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