Integrated circuits from a certain factory pass a particular quality test with p

Question

Integrated circuits from a certain factory pass a particular quality test with probability 0.76. The outcomes of all tests are mutually independent. (a) What is the expected number of tests necessary to find 660 acceptable circuits? (b) Use the central limit theorem to estimate the probability of finding at least 660 acceptable circuits in a batch of 858 circuits. Note that this is a discrete random variable, so don't forget to use "continuity correction". (c) Now use the central limit theorem to calculate the minimum batch size n for finding at least 660 acceptable circuits with probability 0.9 or greater. Note that your answer must be an integer.

Explanation / Answer

from binomial

a)expected number =np=660 ; n=660/0.76 =868.42~869

b)for 858 circuits ; mean =np=858*0.76=652.08

and std deviation =(np(1-p))1/2 =12.51

hence P(X>=660)=1-P(X<=659) =1-P(Z<(659.5-652.08)/12.51)=1-P(Z<0.5931)=1-0.7235=0.2765

c)let n is required test

hence P(X>=660)=1-P(X<=659) =1-P(Z<(659.5-0.76n)/0.427(n)1/2)>0.9

P(Z<(659.5-0.76n)/0.427(n)1/2)<0.1

(659.5-0.76n)/0.427(n)1/2 <-1.28

solving above with permutation and combination

n=890

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