A manufacturing process is designed to produce bolts with a 0.5-in. diameter. On

Question

A manufacturing process is designed to produce bolts with a 0.5-in. diameter. Once each day, a random sample of 36 bolts is selected and the diameters recorded. If the resulting sample mean is less than 0.490 in. or greater than 0.510 in., the process is shut down for adjustment. The standard deviation for diameter is 0.02 in. What is the probability that the manufacturing line will be shut down unnecessarily? (Hint: Find the probability of observing an x in the shutdown range when the true process mean really is 0.5 in. Round your answer to four decimal places.)

Explanation / Answer

mean is 0.5 and s is 0.02

for sample size of 36, the standard error SE=s/sqrt(N)=0.02/sqrt(36)=0.00333

we need to find 1-P(0.49<x<0.51)

z is given as (x-mean)/SE

thus 1-P((0.49-0.5)/0.0033<z<(0.51-0.5)/0.0033)=1-P(-3.03<z<3.03) or 1-(2*P(z<3.03)-1) or 2-2*P(z<3.03)

from normal distribution table we get 2(1-0.9988)=0.0024

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