2. A machine at K&A; Tube & Manufacturing Company produces a certain copper tubi

Question

2. A machine at K&A; Tube & Manufacturing Company produces a certain copper tubing component in a refrigeration unit. The tubing components produced by the manufacturer have a mean diameter of 0.75 inch with a standard deviation of 0.004 inch. The quality-control inspector takes a random sample of 30 components once each week and calculates the mean diameter of these components. if the mean is either less than 0.748 inch or greater than 0.752 inch, the inspector concludes that the machine needs an adjustment. (a) What is the sampling distribution of the sample mean diameter for a random sample of 30 such components? Justify your answer. (b) What is the probability that, based on a random sample of 30 such components, the inspector will conclude that the machine needs an adjustment?

Explanation / Answer

This distribution in normal distribution as sample size is very large as well as most of the diameter will be near to mean diameter and probability of values more than 3 sigma limit are very less. Hence this distribution can taken as normal distribution with following parameter.

mean = 0.75 in

sd = 0.004 in

non-adjustable range = (0.748,0.752)

P( adjustment) = 1 - P(non-adjustment)

P( adjustment) = 1 - P(0.748<x<0.752)

P( adjustment) = 1 - P( (0.748-0.75)/0.004 < z < (0.752-0.75)/0.004))

P( adjustment) = 1 - P(-0.5 < z < 0.5)

P( adjustment) = 1 - [ P(z<0.5) - P(z< -0.5) ]

P( adjustment) = 1 - [0.6915 - 0.3058]

P( adjustment) = 0.6143

Probability of adjustment is 0.6143

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