Chapter 16, Problem 19CAP

(a)

To determine

Find the equation of the regression line.

Answer to Problem 19CAP

The equation of the regression line is, $\widehat{Y}=0.36X+13.17$.

Explanation of Solution

Calculation:

The information given that is the case files to record the age of five criminals and the age of the victim in those cases with the factors age of a criminal (X) predicts the age of the victim (Y) for nonviolent crimes.

The procedure of straight line by using the method of least square is shown below:

• Compute preliminary calculations.
• Calculate the slope $\left(b\right)$.
• Calculate the y-intercept $\left(a\right)$.

The formula for slope is,

$\text{slope}\left(b\right)=\frac{S{S}_{XY}}{S{S}_X}$

Where,

$S{S}_{XY}={\displaystyle \sum \left(X-M_X\right)\left(Y-M_Y\right)}$

$S{S}_X={\displaystyle \sum {\left(X-M_X\right)}^2}$

In the formulas, $M_X$ denotes the mean of factor X, $M_Y$ denotes the mean of factor Y.

The formula for y-intercept is,

$a=M_Y-b{M}_X$

The regression equation is, $\widehat{Y}=bX+a$.

The value of $S{S}_X$:

The value of mean for X is,

$\begin{array}{c}{M}_X=\frac{32+24+28+17+12}{5}\\ =\frac{113}{5}\\ =22.6\end{array}$

X	Y	$\left(X-M_X\right)$	${\left(X-M_X\right)}^2$
32	24	$\left(32-22.6\right)=9.4$	88.36
24	20	$\left(24-22.6\right)=1.4$	1.96
28	25	$\left(28-22.6\right)=5.4$	29.16
17	22	$\left(17-22.6\right)=-5.6$	31.36
12	16	$\left(12-22.6\right)=-10.6$	112.36
			${\displaystyle \sum {\left(X-M_X\right)}^2}=263.2$

Table 1

The value of $S{S}_X$ is 263.2.

The value of $S{S}_{XY}$:

The value of mean for Y is,

$\begin{array}{c}{M}_Y=\frac{24+20+25+22+16}{5}\\ =\frac{107}{5}\\ =21.4\end{array}$

X	Y	$\left(X-M_X\right)$	$\left(X-M_Y\right)$	$\left(X-M_X\right)\left(X-M_Y\right)$
32	24	$\left(32-22.6\right)=9.4$	$\left(24-21.4\right)=2.6$	24.44
24	20	$\left(24-22.6\right)=1.4$	$\left(20-21.4\right)=-1.4$	–1.96
28	25	$\left(28-22.6\right)=5.4$	$\left(25-21.4\right)=3.6$	19.44
17	22	$\left(17-22.6\right)=-5.6$	$\left(22-21.4\right)=0.6$	–3.36
12	16	$\left(12-22.6\right)=-10.6$	$\left(16-21.4\right)=-5.4$	57.24
				${\displaystyle \sum \left(X-M_X\right)\left(X-M_Y\right)}=95.8$

Table 2

The value of $S{S}_{XY}$ is 95.8.

Slope:

Substitute, $S{S}_X=263.2$ and $S{S}_{XY}=95.8$ in slope formula

$\begin{array}{c}\text{slope}\left(b\right)=\frac{95.8}{263.2}\\ =0.364\end{array}$

Hence, the value of slope is 0.364.

Substitute, $M_Y=21.4,M_X=22.6,b=0.364$ in y-intercept formula.

$\begin{array}{c}a=M_Y-b{M}_X\\ =21.4-\left(0.364\right)22.6\\ =21.4-8.2264\\ =13.174\end{array}$

Hence, the value of y-intercept is 13.174.

Substitute, $a=13.174,b=0.364$ in regression equation formula.

$\begin{array}{c}\widehat{Y}=\left(0.364\right)X+13.174\\ =0.36X+13.17\end{array}$

The equation of the regression line is, $\widehat{Y}=0.36X+13.17$.

(b)

To determine

Find the predicted age of a victim of a nonviolent crime when the criminal is 20 years old.

Answer to Problem 19CAP

The predicted age of a victim of a nonviolent crime when the criminal is 20 years old is 20.37 years old.

Explanation of Solution

Calculation:

The given information is that the criminal is 20 years old.

The equation of the regression line is $\widehat{Y}=0.36X+13.17$.

Substitute, $X=20$ in regression line.

$\begin{array}{c}\widehat{Y}=0.36\left(20\right)+13.174\\ =7.2+13.17\\ =20.37\end{array}$

Hence, the predicted age of a victim of a nonviolent crime when the criminal is 20 years old is 20.37 years old.