Chapter 6, Problem 6.25E

a.

To determine

To make: A two-way table showing defendant’s race and death penalty.

Answer to Problem 6.25E

The two-way table of defendant’s race and death penalty is given below:

	Defendant’s Race
	White	Black
Death	19	17
No death	141	149

Explanation of Solution

Given info:

The data show the 326 cases where the defendant was sentenced for committing murder. The results are tabulated below:

White defendant:

	White Victim	Black Victim
Death	19	0
No death	132	9
Total	151	9

Black defendant:

	White Victim	Black Victim
Death	11	6
No death	52	97
Total	63	103

Calculation:

The total number of deaths and no death committed by white defendant is given below:

	White Victim	Black Victim	Total
Death	19	0	19
No death	132	9	141

Table 1

The total number of deaths and no death committed by black defendant is given below:

	White Victim	Black Victim	Total
Death	11	6	17
No death	52	97	149

Table 2

Thus, combing the totals obtained in Table 1 and Table 2 gives the two-way table for defendants and death penalty.

	Defendant’s Race
	White	Black
Death	19	17
No death	141	149

Justification:

The two-way table is obtained by adding the values under each row of the two tables for white and black defendants.

b.

To determine

To show: The Simpson paradox holds good for the given two-way table.

Answer to Problem 6.25E

The Simpson paradox holds good for the given two-way table.

Explanation of Solution

Calculation:

The percentage of black victim given death sentence under white defendant table is calculated as follows:

$\begin{array}{c}\left(\begin{array}{l}\text{Percentage of black victim}\\ \text{given death sentence under}\\ \text{white defendant}\end{array}\right)=\frac{\left(\begin{array}{l}\text{Number of black victims given}\\ \text{death sentence under white defendant}\end{array}\right)}{\left(\begin{array}{l}\text{Total number of black victims}\\ \text{under white defendant}\end{array}\right)}\times 100\\ =\frac{0}{9}\times 100\\ =0\end{array}$

Thus, none of the black victims is sentenced to death under white defendant.

The percentage of black victims given death sentence under black defendant table is calculated as follows:

$\begin{array}{c}\left(\begin{array}{l}\text{Percentage of black victim}\\ \text{given death sentence under}\\ \text{black defendant}\end{array}\right)=\frac{\left(\begin{array}{l}\text{Number of black victims given death}\\ \text{sentence under black defendant}\end{array}\right)}{\left(\begin{array}{l}\text{Total number of black victims}\\ \text{under black defendant}\end{array}\right)}\times 100\\ =\frac{6}{103}\times 100\\ =0.06\times 100\\ =6\end{array}$

Thus, 6% of the black victim is sentenced to death under black defendant.

The percentage of white victim given death sentence under white defendant table is calculated as follows:

$\begin{array}{c}\left(\begin{array}{l}\text{Percentage of white victim}\\ \text{given death sentence under}\\ \text{white defendant}\end{array}\right)=\frac{\left(\begin{array}{l}\text{Number of white victims given death}\\ \text{sentence under white defendant}\end{array}\right)}{\left(\begin{array}{l}\text{Total number of white victim}\\ \text{under white defendant}\end{array}\right)}\times 100\\ =\frac{19}{151}\times 100\\ =0.126\times 100\\ =12.6\%\end{array}$

Thus, 12.6% of the white victim is sentenced to death under white defendant.

The percentage of white victims given death sentence under black defendant table is calculated as follows:

$\begin{array}{c}\left(\begin{array}{l}\text{Percentage of white victim}\\ \text{given death sentence under}\\ \text{black defendant}\end{array}\right)=\frac{\left(\begin{array}{l}\text{Number of white victims given death}\\ \text{sentence under black defendant}\end{array}\right)}{\left(\begin{array}{l}\text{Total number of white victim}\\ \text{under black defendant}\end{array}\right)}\times 100\\ =\frac{11}{63}\times 100\\ =0.175\times 100\\ =17.5\end{array}$

Thus, 17.5% of the white victim is sentenced to death under black defendant.

The percentage of death sentence given to white defendant is calculated as follows:

$\begin{array}{c}\left(\begin{array}{l}\text{Percentage of death}\\ \text{among white defendants}\end{array}\right)=\left(\frac{\text{Number of death for white defendants}}{\text{Total number of white defendants}}\right)\times 100\\ =\frac{19}{160}\times 100\\ =0.119\times 100\\ =11.9\end{array}$

Thus, 11.9% of death sentence was given to white defendant.

The percentage of death sentence given to black defendant is calculated as follows:

$\begin{array}{c}\left(\begin{array}{l}\text{Percentag of death}\\ \text{among black defendants}\end{array}\right)=\left(\frac{\text{Number of death for black defendants}}{\text{Total number of black defendants}}\right)\times 100\\ =\frac{17}{166}\times 100\\ =0.102\times 100\\ =10.2\end{array}$

Thus, 10.2% of death sentence was given to black defendant.

The percentages of deaths under the white defendant and black defendant are given below:

	Defendant’s Race
	White	Black
Death	11.9%	10.2%

The individual percentage of deaths under white victims and black victims is given below:

	White Victim	Black Victim
Death for White defendant	12.6%	0
Death for Black defendant	17.5%	6%

Thus, the Simpson's paradox is proved.2

c.

To determine

To explain: The existence of Simpson’s paradox for the given data to a judge.

Answer to Problem 6.25E

The death penalty was assigned mainly for the cases involving white victims, which is 14% of cases with a white victim and 5.4% of cases with a black victim. Also, most of the white defendants are white victims, and cases having white victims have additional risk of death penalty, white defendants are being assigned the death penalty more overall.

Explanation of Solution

Calculation:

For white defendants, the percentage of white victims is calculated as follows:

$\begin{array}{c}\left(\begin{array}{l}\text{Percentage of white victims}\\ \text{under white defendants}\end{array}\right)=\frac{\left(\begin{array}{l}\text{Number of white victims given death sentence+}\\ \text{Number of white victims given no death sentence}\end{array}\right)}{\text{Total number of white defendants}}\times 100\\ =\left(\frac{19+132}{19+132+0+9}\right)\times 100\\ =\frac{151}{160}\times 100\\ =94.4\end{array}$

Thus, 94.4% of white victims are under white defendants.

For black defendants, the percentage of white victims is calculated as follows:

$\begin{array}{c}\left(\begin{array}{l}\text{Percentage of white victims}\\ \text{under black defendants}\end{array}\right)=\frac{\left(\begin{array}{l}\text{Number of white victims given death sentence+}\\ \text{Number of white victims given no death sentence}\end{array}\right)}{\text{Total number of black defendants}}\times 100\\ =\left(\frac{11+52}{11+52+6+97}\right)\times 100\\ =\frac{63}{166}\times 100\\ =37.95\end{array}$

Thus, 37.95% of white victims are under black defendants.

The overall death penalty involving white and black victims are calculated as follows:

White defendant:

	White Victim	Black Victim
Death	19	0
No death	132	9
Total	151	9

Black defendant:

	White Victim	Black Victim
Death	11	6
No death	52	97
Total	63	103

Adding the corresponding columns of white victim and black victim

	White Victim	Black Victim
Death	$19+11=30$	$0+6=6$
Total	$151+63=214$	$9+103=112$

The percentage of cases involving white victims is given below:

$\begin{array}{c}\left(\begin{array}{l}\text{Percentage of cases}\\ \text{involving white victims}\end{array}\right)\text{ =}\frac{30}{214}\times 100=0.14\times 100\\ =14\%\end{array}$

Thus, the percentage of cases involving white victims is 14%.

The percentage of cases black victims is given below:

$\begin{array}{c}\left(\begin{array}{l}\text{Percentage of cases}\\ \text{involving black victims}\end{array}\right)\text{ =}\frac{6}{112}\times 100=0.054\times 100\\ =5.4\%\end{array}$

Thus, the percentage of cases involving black victims is 5.4%.