Life tests of cutting tools. Refer to the data on life tests of cutting tools, Exercise 11.52

(p. 636).

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 x that was not included in the original sample. That is, the value x = 45 was not part of the sample. However, the value is within the range of x values in the sample, so that 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 x = 100 meters per minute. Since the given value of x is outside the range of the sample x 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% confidence 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?