8.73 York Steel Corporation produces iron rings that are supplied to other compa

Question

8.73 York Steel Corporation produces iron rings that are supplied to other companies. These rings are supposed to have a diameter of 24 inches. The machine that makes these rings does not produce each ring with a diameter of exactly 24 inches. The diameter of each of the rings varies slightly. It is known that when the machine is working properly, the rings made on this machine have a mean diameter of 24 inches. The standard deviation of the diameters of all rings produced on this machine is always equal to .06 inch. The quality control department takes a random sample of 25 such rings every week, calculates the mean of the diameters for these rings, and makes a 99% confidence interval for the population mean. If either the lower limit of this confidence interval is less than 23.975 inches or the upper limit of this confidence interval is greater than 24.025 inches, the machine is stopped and adjusted. A recent such sample of 25 rings produced a mean diameter of 24.015 inches. Based on this sample, can you conclude that the machine needs an adjustment? Explain. Assume that the population distribution is approximately normal.

Explanation / Answer

Confidence Interval
CI = x ± Z a/2 * (sd/ Sqrt(n))
Where,
x = Mean
sd = Standard Deviation
a = 1 - (Confidence Level/100)
Za/2 = Z-table value
CI = Confidence Interval
Mean(x)=24
Standard deviation( sd )=0.06
Sample Size(n)=25
Confidence Interval = [ 24 ± Z a/2 ( 0.06/ Sqrt ( 25) ) ]
= [ 24 - 2.58 * (0.012) , 24 + 2.58 * (0.012) ]
= [ 23.969,24.031 ]

the machine needs an adjustment since it the lower limit is below to
23.975 inches and upper limit is greater than 24.025 inches

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