A manufacturing process is designed to produce bolts with a 0.75-inch diameter.

Question

A manufacturing process is designed to produce bolts with a 0.75-inch diameter. Once each day, a random sample of 36 bolts is selected and the bolt diameters recorded. If the resulting sample mean is less than 0.730 inches or greater than 0.770 inches, the process is shut down for adjustment. The standard deviation for diameter is 0.04 inches. 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 0.75 inches. Round your answer to four decimal places.)

Explanation / Answer

Solution:

Since both 0.73 and 0.77 are exactly 0.02 away from the mean (0.75), use the symmetry of the Normal distrubution to solve for P

P(X < 0.73) = P[z < (0.73 - 0.75) / (0.04 / sqrt 36)

= P(z < -3) = 0.00135 [from Standard Normal table]

Now remember 0.00135 is the solution for P(X < 0.73), but with symmetry, P(X > 0.77) = 0.00135.

So P = 2 x 0.00135 = 0.0027

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