Question

involving a battered woman who had shot and killed her husband. In particular, the researchers manipulated the 30. (Table: Abuse) In a study of simulated juror decision making, Braden-Maguire, Sigal, and Perrino (2005) race of the defendant (African American or Caucasian, and the type of abuse suffered (emotional or physical) The dependent variable was the number of guilty verdicts. (a) Perform the six steps of hypothesis testing using a chi-square test for independence to determine the effects of the type of abuse and the race of the defendant on the number of guilty verdicts. (6) Calculate the effect size for this test and explain its meaning. (c) Calculate the relative likelihood of being found guilty of emotional abuse if African American rather than Caucasian. The data appear in the table. Table: Abuse Emotional Abuse 19 Physical Abuse 6 African American Caucasian 16 7

Answer 1

(a)

The following cross tabulation have been provided. The row and column total have been calculated and they are shown below:

	Emotional abuse	Physical abuse	Total
African American	19	6	25
Caucasian	16	7	23
Total	35	13	48

The expected values are computed in terms of row and column totals. The formula is  
Eij Ri X Ci T
​​, where $R_i$ ​ corresponds to the total sum of elements in row i, $C_j​$ corresponds to the total sum of elements in column j, and T is the grand total. The table below shows the calculations to obtain the table with expected values:

Expected Values	Emotional abuse	Physical abuse	Total
African American	35 x 25 18.229 4 8	$\frac{ 13 \times 25}{ 48} = 6.771$	25
Caucasian	$\frac{ 35 \times 23}{ 48} = 16.771$	$\frac{ 13 \times 23}{ 48} = 6.229$	23
Total	35	13	48

Based on the observed and expected values, the squared distances can be computed according to the following formula: . The table with squared distances is shown below:

Squared Distances	Emotional abuse	Physical abuse
African American	$\frac{(19 - 18.229)^2}{ 18.229} = 0.033$	$\frac{(6 - 6.771)^2}{ 6.771} = 0.088$
Caucasian	$\frac{(16 - 16.771)^2}{ 16.771} = 0.035$	$\frac{(7 - 6.229)^2}{ 6.229} = 0.095$

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

$H_0​$: The two variables are independent

$H_a​:$ :The two variables are dependent

This corresponds to a Chi-Square test of independence.

Rejection Region

Based on the information provided, the significance level is
g= 0,05
, the number of degrees of freedom is $df = (2 - 1) \times (2 - 1) = 1$ , so then the rejection region for this test is $R = \{\chi^2: \chi^2 > 3.841\}.$

Test Statistics

The Chi-Squared statistic is computed as follows:

$\chi^2 = \sum_{i=1}^n \frac{(O_{ij}-E_{ij})^2}{E_{ij}}= 0.033+0.035+0.088+0.095 = 0.251$

Decision about the null hypothesis

Since it is observed that $\chi^2 = 0.251 \le \chi_c^2 = 3.841$ , it is then concluded that the null hypothesis is not rejected.

Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is NOT enough evidence to claim that the two variables are dependent, at the 0.05 significance level.

(b) effect size for this test is given by the phi coefficient formula $\phi =\sqrt{\frac{\chi ^2}{n}}$

$\phi =\sqrt{\frac{0.251}{48}}=\sqrt{0.005229}=0.0723$

since value of phi is less than 0.1 hence there is a small effect.

(c)

relative liklihood of being found guilty of emotional abuse of African American is given by

$\frac{19}{48}=0.3958$