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PROBLEM 8.3 The diameter of bushings turned out by a manufacturing process is a normally distributed random variable with a mean of 4.035 mm and a...
PROBLEM 8.3The diameter of bushings turned out by a manufacturing process is a normally distributed random variable with a mean of 4.035 mm and a standard deviation of 0.005 mm. The inspection procedure requires a sample of 25 bushings once an hour.(a) Within what range should 95 percent of the bushing diameters fall? Let's collect our data:x-ave = 4.035 mmsigma = 0.005 mm(sample size)confidence = 95%For this one we need to remember we are dealing with items, not mean values. Our common rule is about 68% of the values will lie within 1 standard deviation and 95% will lie within 2 standard deviations.Range = [x-ave - 2*sigma, x-ave + 2*sigma]Range = [4.035 - 2*0.005, 4.035 + 2*0.005]Range = [4.035 - 0.010, 4.035 + 0.010]Range = [4.025, 4.045](b) Within what range should 95 percent of the sample means fall?This one involves sample means so we use the confidence interval. Our formula for the confidence interval follows:x-ave +/- z * sigma / sqrt(n)For z we can look at the z tables OR the t tables (this is a nice trick).From the z table… the 95% confidence means 95% / 2 or 47.5% of the confidence lies on either side of the mean value. In the z table we are going to find the closest match to 0.475. Here is a quick guide for the conversion I just did: consider the 47.5% to be pennies and the 0.475 to be in our decimal dollars notation. Looking up the value we find 0.475 located in the column headed by 0.06 and the row headed by 1.9. Add these numbers together to give you the z value of 1.96.z = 1.96Filling in the values we have4.035 +/- 1.96 * 0.005 / sqrt(5)4.035 +/- 1.96 * 0.0014.035 +/- 0.002Range = [4.035-0.002, 4.035+0.002]Range = [4.033, 4.037](c) What conclusion would you reach if you saw a sample mean of 4.020? A sample meanof 4.055?For 4.020: it is within the range so I would conclude there is no problem.For 4.020: the mean lies below the range so I would conclude we have a system out of control.
