# solution

Life tests of cutting tools. Refer to the data saved in the file, Exercise 11.44 (p. 663).

a. Use a 90% confidence interval to estimate the mean useful life of a brand A cutting tool when the cutting speed is 45 meters per minute. Repeat for brand B. Compare the widths of the two intervals and comment on the reasons for any difference.

b. Use a 90% prediction interval to predict the useful life of a brand A cutting tool when the cutting speed is 45 meters per minute. Repeat for brand B. Compare the widths of the two intervals with each other and with the two intervals you calculated in part a. Comment on the reasons for any differences.

c. Note that the estimation and prediction you performed in parts a and b were for a value of Â that was not included in the original sampleâ€”that is, the value Â = 45 was not part of the sample. However, the value is within the range of -values in the sample, so the regression model spans the x-value for which the estimation and prediction were made. In such situations, estimation and prediction represent interpolations. Suppose you were asked to predict the useful life of a brand A cutting tool for a cutting speed of Â = 100 meters per minute. Because the given value of x is outside the range of the sample -values, the prediction is an example of extrapolation. Predict the useful life of a brand A cutting tool that is operated at 100 meters per minute and construct a 95% prediction interval for the actual useful life of the tool. What additional assumption do you have to make in order to ensure the validity of an extrapolation?

Exercise 11.44

Life tests of cutting tools. To improve the quality of the output of any production process, it is necessary first to understand the capabilities of the process (Gitlow, Quality Management, 2004). In a particular manufacturing process, the useful life of a cutting tool is related to the speed at which the tool is operated. The data in the table below were derived from life tests for the two different brands of cutting tools currently used in the production process. For which brand would you feel more confident in using the least squares line to predict useful life for a given cutting speed? Explain.

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