Question

You are in charge of a manufacturing facility that produces an automotive part (steel shaft for the gearbox) with an acceptable dimension of 2.5±0.05 inches in diameter. Note that, the most desirable product is one with diameter of exactly 2.5 inches. Two vendors are trying to sell you their machines for the shaft-machining task. You have to select a vendor.You asked both vendors to supply data on the machining accuracy of their machines for the given task. Both vendors machined 100 shafts, collected data, plotted histograms, fitted the histograms with normal distributions and supplied you with their findings. Let X= diameter in inches of the gearbox shaft Vendor A: X has a normal distribution with mean 2.48 and variance 0.001 Vendor B: X has a normal distribution with mean 2.51 and variance 0.002 Answer the following: Examine the scenario as described and discuss what additional information would be helpful to make this decision and why that information would be useful. (50 words) Assuming that you have to select a vendor using only the given information, develop and discuss your approach (give all details including quantitative justification). Your response to this question must be directed to the audience described below. (200 words or less)

Answer 1

It would be useful to get metrics like distribution quantiles, range or median. Also the most important aspect would be to get distribution fit statistics such as goodness of fit for normal distribution being fitted to the data in addition to Q-Q plots for normal distribution compared with the data.

A has mean of 2.48 and variance of 0.001 and hence sd 0.032 ( sd : standard deviation). B has mean of 2.51 and variance of 0.002 and hence sd 0.045. Since the means of both are similar in terms of magnitude of difference from the desired level of 2.50, sd plays a pivotal role here. Since 0.045 is quite higher than 0.032, B will have a slight preference. If we compute the tolerance intervals, that is the intervals based on the normal distribution x-sigma limits where x=1,2,3,4,5,6 we see as the x increases the interval widens but the interval width or variability is more in case of A than in B. The overall mean of the tolerance intervals stay quite close to the desired level of 2.5. Here, in simpler terms the issue of accuracy or unbiasedness is similar for both A and B with not a significant difference. However, the precision, consistency or variability of A and B are quite different. The better precision of A makes it a more preferred choice over B. Hence the vendor of choice with just the given information and no additional information would be A.