Solved: Forensic scientists are often interested in making

Chapter 12, Problem 70E

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Forensic scientists are often interested in making a measurement of some sort on a body (alive or dead) and then using that as a basis for inferring something about the age of the body. Consider the accompanying data on age (yr) and % D-aspertic acid (hereafter %DAA) from a particular tooth (“An Improved Method for Age at Death Determination from the Measurements of D-Aspertic Acid in Dental Collagen,” Archaeometry, 1990: 61–70.)

                                   

Suppose a tooth from another individual has 2.01% DAA. Might it be the case that the individual is younger than 22 ? This question was relevant to whether or not the individual could receive a life sentence for murder.

A seemingly sensible strategy is to regress age on %DAA and then compute a PI for age when %DAA = 2.01. However, it is more natural here to regard age as the independent variable x and %DAA as the dependent variable y, so the regression model is % DAA \(=\beta_0+\beta_1 x+\epsilon\). After estimating the regression coefficients, we can substitute \(y^*=2.01\) into the estimated equation and then solve for a prediction of age \(\hat{x}\). This "inverse" use of the regression line is called "calibration." A PI for age with prediction level approximately \(100(1-\alpha) \%\) is \(\hat{x} \pm t_{\alpha / 2, n-2} \cdot S E\) where

                    \(S E=\frac{s}{\hat{\beta}_1}\left\{1+\frac{1}{n}+\frac{(\hat{x}-\bar{x})^2}{S_{x x}}\right\}^{1 / 2}\)

Calculate this PI for \(y^*=2.01\) and then address the question posed earlier.