Question

Suppose a shipment of 400 components contains 68 defective and 332 non-defective computer components. From the shipment you take a random sample of 25. When sampling with replacement (so that the p = probability of success does not change), note that a success in this case is selecting a defective part. In our sample of 25 we reject the entire shipment if X>5. What is the probability of rejecting the entire shipment?

Answer 1

Answer:

0.2425 or 24.25%

Step-by-step explanation:

If the shipment of 400 components contains 68 defective and, we can estimate the probability of selecting a defective components as

68/400 = 0.17

Now, we have a binomial experiment of 25 trials and we want to compute the probability that more than 5 trials are “success” (defective component)

P(X>5) = 1 - P(X≤ 5)

Since  

1 - P(X≤ 5) = 1 - (P(X=0) + P(X=1) + P(X=3) + P(X=4) + P(X=5) =  

$1-(\displaystyle\binom{25}{0}(0.17)^0(1-0.17)^{25}+\displaystyle\binom{25}{1}(0.17)^1(1-0.17)^{24}+\displaystyle\binom{25}{2}(0.17)^2(1-0.17)^{23}+\\\\+\displaystyle\binom{25}{3}(0.17)^3(1-0.17)^{22}+\displaystyle\binom{25}{4}(0.17)^4(1-0.17)^{21}+\displaystyle\binom{25}{5}(0.17)^5(1-0.17)^{20})=\\\\\Rightarrow \boxed{P(X>5)=0.2425}$