5. Using four decimal digit accuracy, explain if the respective distribution app

Question

5. Using four decimal digit accuracy, explain if the respective distribution applies and, if so, determine the probability that at least 15 and at most 17 of 2500 randomly tested micro circuits from a shipment of 500,000 will fail if historically on average 1 in 125 failed.

5.1 If it applies, use a Binominal distribution

5.2 If it applies, use a Poisson distribution

5.3 If it applies, use a Normal distribution

5.4 Explain why statistically the preceding three results differ if they apply and differ

Explanation / Answer

Here we have the probability P as 1/125 which is fixed in each and every case. Here the number of trials is finite as 2500. The trials are independent of each other so this is the case of Binomial Distribution. But here the sample size is large (>30) and np and nq values are greater than 5 so normal approximation to binomial will be the most applicable here.

n = 2500

P = 1/125

np = 2500 * (1/125) = 20 > 5

nq = 2500 * ((1 – (1/125) = 2480 > 5

P ( 15 <= x <= 17)

Mean = np = 20

Standard deviation = sqrt (npq) = sqrt (2500*(1/125)*(124/125)) = 4.4542

P (14.5 < x < 17.5)

z1 = (14.5 – 20)/4.4542 = -1.23

z2 = (17.5 – 20)/4.4542 = -0.56

P (-1.23 < z < -0.56)

P (z<-0.56) – P (z<-1.23)

= 0.2877 – 0.1093

= 0.1784

Answer: 0.1784

5.3 Using normal distribution we get the probability as 0.1784

5.4 The assumptions will not satisfy if we use Simple Binomial or Poisson distribution so the obtained answers will differ from the actual one.

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