Chapter 11.6, Problem 11.96ACA

Life tests of cutting tools Refer to the data saved in the file. Exercise 11.44 (p 639).

1. 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 8 Compare the widths of the two intervals and comment on the reasons for any difference.
2. 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,
3. 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 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. Because 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% 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?

In the preceding sections, we have presented the basic elements necessary to fit and use a straight-line regression model In this section, we will assemble these elements by applying them in an example with the aid of a computer.

Suppose a fire insurance company wants to relate the amount of fire damage in major residential fires to the distance between the burning house and the nearest fire station. The study is to be conducted in a large suburb of a major city; a sample of 15 recent fires in this suburb is selected The amount of damage, y, and the distance between the fire and the nearest fire station, x, are recorded for each fire. The results are shown in Table 11.5 and saved in the RFIRES file

Table 11.5 Fire Damage Data

Distance from Fire Station, x (miles)	Fire Damage, y (thousands of dollars)
3.4	26.2
1.8	17.8
46	313
23	23.1
3.1	27.5
5.5	360
.7	14.1
3.0	22.3
2.6	19.6
4.3	31 3
2 1	240
1.1	17.3
6.1	432
48	36 4
38	26 1