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In July 2013 , Jeff Witmer obtained a data set from the Tampa Bay Times after the Zimmerman case was decided. Zimmerman shot and killed Trayvon Martin (an unarmed black teenager) and was acquitted. The data set concerns "stand your ground" cases with male defendants. Some of these were fatal attacks, and some were not fatal. Many of those charged used guns, but some used various kinds of knives or other methods. $$ \begin{array}{|lccc|} \hline & & \text { Accused } \\ \hline & \text { Nonwhite } & \text { White } & \text { All } \\ \hline \text { Not Convicted } & 48 & 80 & 128 \\ \hline \text { Convicted } & 19 & 38 & 57 \\ \hline \text { All } & 67 & 118 & 185 \\ \hline \end{array} $$ a. What percentage of the nonwhite defendants were convicted? b. What percentage of the white defendants were convicted? c. Test the hypothesis that conviction is independent of race at the \(0.05\) level. Assume you have a random sample.

Step by step solution

01

Percentage of nonwhite defendants convicted

To calculate the percentage of nonwhite defendants convicted, take the number of nonwhite convictions and divide by total nonwhite defendants. That is, \( \frac{19}{67} \times 100 \%\).

02

Percentage of white defendants convicted

To calculate the percentage of white defendants convicted, take the number of white convictions and divide by total white defendants. That is, \( \frac{38}{118} \times 100 \%\).

03

Hypothesis Testing

In considering the hypothesis that conviction is independent of race, first establish null and alternate hypotheses. The null hypothesis (H0) is that conviction rates are independent of race. The alternative hypothesis (Ha) is that conviction rates are dependent on race. Use a chi-square test to determine whether to reject the null hypothesis or fail to reject it. The test statistic \( \chi^2 \) is calculated as: \( \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \) where \( O_i \) are the observed frequencies (from the table) and \( E_i \) are the expected frequencies if H0 is true. If the calculated \( \chi^2 \) is greater than the critical value from the chi-square distribution table at 0.05 significance level, the null hypothesis is rejected.

04

Calculate Expected Frequencies

Calculate expected frequencies for each cell by multiplying the column total by row total and divide by overall total. Calculation formula: \( E_i = \frac{(row_i \ total) \times (column_i \ total)}{Overall \ total} \). Calculate for each cell.

05

Calculate the Chi-square Test Statistic

Substitute the observed and expected frequencies into the chi-square formula to find the test statistic. Add all these values to get the total \( \chi^2 \) statistic.

06

Compare with Critical Value

Compare the calculated \( \chi^2 \) with the critical value from the chi-square distribution table at 0.05 significance level (1 degree of freedom since it's a 2x2 table). If the calculated \( \chi^2 \) exceeds the critical value, reject the null hypothesis.

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Key Concepts

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

Chi-square Test

The chi-square test is a wonderful tool in statistics used to determine if there's a significant association between two categorical variables. In the context of this exercise, we use the chi-square test to explore whether conviction rates are independent of race. Let's break it down further.

• We start by establishing two hypotheses: the null hypothesis (H0) asserts that the conviction rates are the same regardless of race, indicating independence. Conversely, the alternate hypothesis (Ha) implies a dependency between conviction rates and race.
• The formula used to calculate the chi-square test statistic (χ²) is: \(χ² = \sum \frac{(O_i - E_i)^2}{E_i}\). Here, \(O_i\) denotes the observed frequencies from our table, and \(E_i\) represents the frequencies expected under the null hypothesis.
• These expected frequencies are found using: \(E_i = \frac{(row_i \ total) \times (column_i \ total)}{Overall \ total}\).
• For this test to be valid, the expected frequency in each cell should be at least 5.

Finally, if the calculated chi-square statistic is greater than the critical value at the 0.05 significance level with one degree of freedom (since we have a 2x2 table), we reject the null hypothesis. If it's less, we fail to reject the null hypothesis. Thus, the chi-square test allows us to deduce whether conviction rates are independent of race.

Conviction Rates

Conviction rates refer to the percentage of defendants in a particular category who are convicted. In this exercise, we are examining the conviction rates separately for nonwhite and white defendants.

• For nonwhite defendants, we calculate the conviction rate by dividing the number of nonwhite convictions (19) by the total number of nonwhite defendants (67) and then multiplying by 100 to convert it to a percentage. This gives approximately 28.4% conviction rate.
• Similarly, for white defendants, the conviction rate is the number of white convictions (38) divided by the total number of white defendants (118), again multiplied by 100, equating to about 32.2%.

These rates provide a direct look at the outcomes of "stand your ground" cases by race. Higher corresponding rates indicate a greater proportion of convictions within those categories. By comparing these rates, we can explore potential discrepancies or patterns in conviction outcomes across different racial groups.

Statistical Independence

Statistical independence between two variables implies that the occurrence of one variable does not affect the probability of the other variable occurring. In hypothesis testing, especially with chi-square tests, this concept is crucial.

When examining conviction rates by race:

• If we determine statistical independence, it implies that the race of a defendant is not related to the likelihood of conviction.
• This means that, statistically, whether a defendant is nonwhite or white should not alter their probability of being convicted if race and conviction are truly independent.
• On the other hand, if we find a significant relationship, it means there might be a dependency, indicating possible disparity in convictions based on race.

Testing for statistical independence helps us explore deeper into whether external factors, such as racial biases, might influence judicial outcomes. It's a key concept because if we fail to establish independence statistically, it could signal systemic issues in the application of justice.