424 Chapter 9: Regression
(d) For a nominal 1-inch screw, find a 90 percent prediction interval for its actual
length.
(e) Plot the standardized residuals.
(f ) Do the residuals in part (e) indicate any flaw in the regression model?
(g) Determine the index of fit.
28.Glass plays a key role in criminal investigations, because criminal activity often
results in the breakage of windows and other glass objects. Since glass fragments
often lodge in the clothing of the criminal, it is of great importance to be able
to identify such fragments as originating at the scene of the crime. Two physical
properties of glass that are useful for identification purposes are its refractive index,
which is relatively easy to measure, and its density, which is much more difficult
to measure. The exact measurement of density is, however, greatly facilitated if
one has a good estimate of this value before setting up the laboratory experiment
needed to determine it exactly. Thus, it would be quite useful if one could use the
refractive index of a glass fragment to estimate its density.
The following data relate the refractive index to the density for 18 pieces of
glass.Refractive Index Density Refractive Index Density
1.5139 2.4801 1.5161 2.4843
1.5153 2.4819 1.5165 2.4858
1.5155 2.4791 1.5178 2.4950
1.5155 2.4796 1.5181 2.4922
1.5156 2.4773 1.5191 2.5035
1.5157 2.4811 1.5227 2.5086
1.5158 2.4765 1.5227 2.5117
1.5159 2.4781 1.5232 2.5146
1.5160 2.4909 1.5253 2.5187(a) Predict the density of a piece of glass with a refractive index 1.52.
(b) Determine an interval that, with 95 percent confidence, will contain the
density of the glass in part (a).
29.The regression modelY=βx+e, e∼N(0,σ^2 )is called regression through the origin since it presupposes that the expected
response corresponding to the input levelx=0 is equal to 0. Suppose that
(xi,Yi),i=1,...,nis a data set from this model.
(a) Determine the least squares estimatorBofβ.
(b) What is the distribution ofB?