To improve the quality of the output of any production process, it is necessary first to understand the capabilities of the process (Deming, Out of the Crisis, 1982). In a particular manufacturing process, the useful life of a cutting tool is linearly related to the speed at which the tool is operated. The data in the accompanying table were derived from life tests for the two different brands of cutting tools currently used in the production process.

a. Fit the model, E(y) = β_{o} + β_{1}x, to the data for brand A, where y = useful life and x = cutting speed.

b. Repeat part a for brand B.

c. 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.

d. 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 to each other and to the two intervals you calculated in part c. Comment on the reasons for any differences.

e. Note that the estimation and prediction you performed in parts c and d 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?