Integrated circuits from a certain factory pass a certain quality test with prob
ID: 3695062 • Letter: I
Question
Integrated circuits from a certain factory pass a certain quality test with probability 0.8. The outcomes of all tests are mutually independent. What is the expected number of tests necessary to find 500 acceptable circuits Use the central limit theorem to estimate the probability of finding 500 acceptable circuits in a batch of 600 circuits. Use Matlab to calculate the actual probability of finding 500 acceptable circuits in a batch of 600 circuits. Use the central limit theorem to calculate the minimum batch size for finding 500 acceptable circuits with probability 0.9 or greater.
Explanation / Answer
Answer:
A) Define X to be the number of tests necessary to find 500 acceptable circuits. X is a negative binomial(500,0.8) random variable.
E[X] = 500/0.8 = 625.
(B) Define Yn to be the acceptable circuits in a batch of n circuits. Yn is a binomial(n,0.8) random variable. Notice that the sum of n iid Bernoulli(p) random variable is a binomial(n,p) random variables. There- fore, we can treat Yn as the sum of n iid Bernoulli(0.8) random vari- ables. We can apply Central Limit Theorem to approximate Yn. We
have
E[Yn] = 0.8n, Var[X] = n · 0.8 · 0.2 = 0.16n.
Therefore, Yn is approximately N(0.8n, 0.16n).
Now we need to calculate P[Y600 500]. From the above conclusion
we know Y600 is approximately N(480, 96). Therefore,
P[Y600 500] = P
Y600 480
96
500 480
96
1[2.04] = 0.02068.
(D) We need to find the smallest n such that P[Yn 500] 0.9. This means
P
Yn 0.8n
0.16n
500 0.8n
0.16n
= 1
500 0.8n
0.16n
0.9.
From Table 3.1 we know (1.28) = 0.9, which means (1.28) = 0.1. Therefore, we have
500 0.8n
0.16n
0.1
500 0.8n
0.16n
1.28.
0.8n 0.512
n 500 0
Solving this inequality, we have
n 24.68205 n 609.2
Therefore, the smallest value of n is 610.
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