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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 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 normal.

Step by step solution

01

Identify given information

The given information for this problem includes: the sample mean (\(24.015\) inches), the population mean (\(24\) inches), the standard deviation (\(.06\) inch), the sample size (25 rings), and the confidence level (\(99\%\)). The target mean range is \(23.975\) to \(24.025\) inches.

02

Calculate the Standard Error

The standard error of the mean is calculated by dividing the standard deviation (\(.06\)) by the square root of the sample size (25). This results in a standard error of \(0.012\).

03

Calculate the Confidence Interval

The \(99\%\) confidence level corresponds to a z-score of approximately \(2.576\). We calculate the confidence interval by multiplying the standard error by the z-score. Here, we get \(2.576 * 0.012 = 0.030912\). So, our confidence interval is \(24.015 \pm 0.030912\), which gives us a range of approximately \(23.984088\) to \(24.045912\) inches.

04

Compare and Conclude

Comparing this confidence interval with the accepted range, \(23.975\) to \(24.025\), we can see that the upper limit of our confidence interval, \(24.045912\), is greater than the upper limit of the accepted range, \(24.025\). Therefore, the machine does need an adjustment.

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