91Ó°ÊÓ

A study of the death penalty in Kentucky reported the results shown in the table. (Source: Data from \(\mathrm{T}\). Keil and \(\mathrm{G}\). Vito, Amer. \(J\). Criminal Justice, vol. \(20,1995,\) pp. \(17-36 .)\) a. Find and compare the percentage of white defendants with the percentage of black defendants who received the death penalty, when the victim was (i) white and (ii) black. b. In the analysis in part a, identify the response variable, explanatory variable, and control variable. c. Construct the summary \(2 \times 2\) table that ignores, rather than controls, victim's race. Compare the overall percentages of white defendants and black defendants who got the death penalty (ignoring, rather than controlling, victim's race). Compare to part a. d. Do these data satisfy Simpson's paradox? If not, explain why not. If so, explain what is responsible for Simpson's paradox occurring. e. Explain, without doing the calculations, how you could inferentially compare the proportions of white and black defendants who get the death penalty (i) ignoring victim's race and (ii) controlling for victim's race. $$\begin{array}{llrrr}\hline \begin{array}{l} \text { Victim's } \\\\\text { Race } \end{array} & \begin{array}{l} \text { Defendant's } \\\\\text { Race } \end{array} & \begin{array}{l}\text { Death Penalty } \\\\\text { Yes } \end{array} & \begin{array}{r} \text { No } \end{array} & \text { Total } \\ \hline \text { White } & \text { White } & 31 & 360 & 391 \\ & \text { Black } & 7 & 50 & 57 \\ \text { Black } & \text { White } & 0 & 18 & 18 \\ & \text { Black } & 2 & 106 & 108 \\\\\hline\end{array}$$

Step by step solution

01

Find Percentages for White Victims

Calculate the percentage of white defendants who received the death penalty when the victim was white. This is given by \( \frac{31}{391} \times 100 \approx 7.93\% \). For black defendants with white victims, the percentage is \( \frac{7}{57} \times 100 \approx 12.28\% \).

02

Find Percentages for Black Victims

Calculate the percentage of white defendants who received the death penalty when the victim was black. This is \( \frac{0}{18} \times 100 = 0\% \). For black defendants with black victims, the percentage is \( \frac{2}{108} \times 100 \approx 1.85\% \).

03

Identify Variables

The response variable is whether the death penalty was given. The explanatory variable is the defendant's race. The control variable is the victim's race.

04

Construct 2x2 Table Ignoring Victim's Race

Sum up the cases ignoring victim's race. White defendants: \(31 + 0 = 31\) received death penalty, total \(391 + 18 = 409\), so percentage \( \frac{31}{409} \times 100 \approx 7.58\% \). Black defendants: \(7 + 2 = 9\) received death penalty, total \(57 + 108 = 165\), so percentage \( \frac{9}{165} \times 100 \approx 5.45\% \).

05

Compare to Part a

Compare percentages calculated in parts (i) and (ii) with those in Step 4. When victim's race is ignored, white defendants show a death penalty percentage of 7.58%, whereas black defendants show 5.45%. Controlled percentages show disparity.

06

Evaluate for Simpson's Paradox

Simpson's paradox occurs when a trend appears in different groups of data but reverses when these groups are combined. Here, the controlled percentages for black defendants being penalized are higher with white victims, but overall, whites seem more penalized when ignoring race, indicating a potential paradox.

07

Inference Comparison Without Calculations

To compare proportions inferentially, statistical tests like chi-squared tests would be conducted. Test (i) without considering race would analyze raw percentages of defendant races receiving the penalty, whereas test (ii) would compare these within race-based victim contexts, reducing confounding factors.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simpson's Paradox: Understanding the Phenomenon

Simpson's Paradox is a fascinating and sometimes confusing occurrence in the field of statistics. It arises when a trend clearly exists in several different groups of data, but reverses or disappears when these groups are combined. This can often lead to misleading interpretations if not properly understood or identified.
To put it simply, a relationship observed within distinct data groups can vanish, or even appear to be opposite, when the groups are aggregated. Hence, it's crucial to control for background variables when analyzing data to avoid falling into the trap of Simpson's Paradox.

• For example, if you have data on treatment success rates for different demographics and see a trend, combining these without considering demographic influence might reverse the observed trend.

In the context of the Kentucky death penalty study, we notice when analyzing defendant race separately for different victim races, the trend suggests one outcome. But when victim race isn't considered, the perspective changes, suggesting the possible presence of Simpson's Paradox.

Response Variable: Defining the Focus

In any statistical analysis, defining the response variable is essential as it represents the outcome of interest or the effect which researchers are attempting to understand or predict. In this particular exercise, the response variable is whether the death penalty was given.
This binary outcome reflects the main result being studied - yes or no.

• Understanding the response variable helps guide the analysis, as all other variables are considered in terms of how they relate to this specific outcome.

In a study on penalties, what happens to the defendant (receiving a death penalty or not) is a crucial factor to observe and will dictate the nature of subsequent analysis.

Explanatory Variable: The Influencer

The explanatory variable, sometimes known as the predictor or independent variable, is the one that is believed to have an effect on the response variable. It may explain variations in the observed data.
In the Kentucky study, the explanatory variable is the defendant's race. This variable is considered because it potentially influences the likelihood of receiving the death penalty, the response variable.

• Explanatory variables are critical in understanding how different attributes of data points could impact the main outcome.

By focusing on the defendant's race, the study can explore how demographic factors may correlate with the probability of receiving the death penalty, leading to insights about systemic biases or disparities.

Control Variable: Mitigating Confounding Effects

When performing an analysis, control variables are employed to mitigate the confounding effects and focus on the primary relationship of interest. They help isolate the true effect of the explanatory variable on the response variable by accounting for other influencing factors.

• In this study, the control variable is the victim's race.

By controlling for the race of the victim, the analysis distinguishes the effect of the defendant's race on the death penalty outcome without the results being skewed by other variations that could lead to misinterpretation.
Properly identifying and accounting for control variables is essential, as they help clarify the primary relationships being examined. Ignoring them could result in misleading conclusions, like those leading to surprises in Simpson's Paradox.

Chi-Squared Test: A Tool for Comparison

The Chi-Squared test is a statistical tool used to determine if there is a significant association between two categorical variables. It helps determine whether observed data deviate from what is expected under a specific hypothesis, usually the null hypothesis that assumes no association.
In the context of this study, the Chi-Squared test could be used to inferentially compare the proportions of white and black defendants who receive the death penalty.

• By utilizing this test, one can statistically evaluate whether any observed differences in death penalty rates between racial groups are due to chance or if they indicate a real association.

This method is particularly useful when comparing categorical datasets, like those in this study, and helps to establish statistical significance in the observed outcomes across different groups.