Creative Robotics recommends that machine shop shut down anautomatic lathe for c

Question

Creative Robotics recommends that machine shop shut down anautomatic lathe for corrective maintenance whenever a sample of theparts it produces has an average diameter greater than 2.01 inchesor smaller than 1.99 inches. The lathe is designed to produce partswith a mean of diameter of 2.00 inches, and the sample averageshave a standard deviation of .005 inch. Assume that the normaldistribution applies.

(a) What is the probability that the lathe is shut down whenoperating as designed, with = 2.00 inches ?

(b) If the lathe begins to produce parts that on the average aretoo wide, with = 2.02 inches what is the probability thatthe lathe will continue to operate ?

Explanation / Answer

a)      Given that the average follows anormal distribution with = 2 and = 0.005

Let <x> represent the average of the sample .

P(shut down) = 1- P(1.99< <x>  > 2.01)

Calculating the z scores:

For 1.99 z = (1.99 – 2)/0.005 = -2

For 2.01 z = (2.01 – 2)/0.005 = 2

P(shut down) = 1- P(-2 < z > 2)

From normal tables P(-2< z > 2) = 0.8920

P(shut down) = 10.79%

b)      Given that the average follows anormal distribution with = 2.02 and = 0.005

Let <x> represent the average of the sample .

P(not shut down) = P(1.99< <x>  > 2.01)

Calculating the z scores:

For 1.99 z = (1.99 – 2.02)/0.005 = -6

For 2.01 z = (2.01 – 2)/0.005 = -2

P(not shut down) =P(-6 < z > -2)

From normal tables P(-6 < z > -2) = 2.27%

Hope this helps. Feel free to ask any clarification

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