Twenty-five percent of all automobiles undergoing an emissions inspection at a c

Question

Twenty-five percent of all automobiles undergoing an emissions inspection at a certain inspection station fail the inspection. (Round all answers to three decimal places.)

(a) Among 18 randomly selected cars, what is the probability that at most 6 fail the inspection?


(b) Among 18 randomly selected cars, what is the probability that between 6 and 12 (inclusive) fail to pass inspection?


(c) Among 25 randomly selected cars, what is the mean value of the number that pass inspection, and what is the standard deviation of the number that pass inspection?
Mean =  
Standard deviation =  

(d) What is the probability that among 25 randomly selected cars, the number that pass is within 1 standard deviation of the mean value?

Explanation / Answer

Answer to the questions is as below:

p ( fail test) = .25

a. P(X<=6) = P(X=0,1..6) = 18C0(.25^0)*(.75^18) +...+18C6.25^6)*(.75^12) = 0.8610

b. P(6<=X<=12) = P(X=6,7,8,9,10,11,12) = 0.9999-0.71745 = .28245

c. Mean = np = 25*.25 = 6.25
Standard deviation = sqrt(npq) = sqrt(25*.25*.75) = 2.165

d. 1 deviation, 6.25 +/- 2.165, The number that are within 1 standard deviation of the mean in a normal distribution, which I think is assumed here, is always 68%

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