A study conducted in Charlotte, North Carolina, tested the effectiveness of three police responses to spouse abuse: (1) advise and possibly separate the couple, (2) issue a citation to the offender, and (3) arrest the offender. Police officers were trained to recognize eligible cases. When presented with an eligible case, a police officer called the dispatcher, who would randomly assign one of the three available treatments to be administered. There were a total of 650 cases in the study. Each case was classified according to whether the abuser was subsequently arrested within six months of the original incident.

(a) Explain the purpose of the random assignment in the design of this study.

(b) Construct a well-labeled graph that is suitable for comparing the effectiveness of the three treatments.

(c) We want to use these data to perform a test of$H_0:$$p_1=p_2=p_3$where$p_i=$

the true proportion of spouse abusers like the ones in this study who would be arrested again within six months after receiving treatment \(i\). State an appropriate alternative hypothesis.

(d) Assume that all the conditions for performing the test in part (b) are met. The test yields ${\chi }^2=5.063$and a P-value of $0.0796$Interpret this P-value in context. What conclusion should we draw from the study?

A study done in Charlotte, North Carolina, examined the effectiveness of three police responses to marital abuse, as stated in the question. The study included a total of 650 patients. Each case was categorized based on whether or not the abuser was apprehended within six months of the initial event.

We need to clarify why the random assignment was used in this study's design.

Let's consider the given table:

$\begin{array}{|ccc|}\hline \text{Treatment}& \text{No subsequent arrest}& \text{Subsequent arrest}\\ \text{Advise and separate}& 187& 25\\ \text{Citation}& 181& 43\\ \text{Arrest}& 175& 39\\ \hline\end{array}$

Random assignment was used in this study to include all of the factors and eliminate the effect of any variables that were not included in the study.

Given in the question that,

$\begin{array}{|ccc|}\hline \text{Treatment}& \text{No subsequent arrest}& \text{Subsequent arrest}\\ \text{Advise and separate}& 187& 25\\ \text{Citation}& 181& 43\\ \text{Arrest}& 175& 39\\ \hline\end{array}$

We need to construct a well-labeled graph for comparing the effectiveness of the three treatments.

The diagram is made in the following way:

We have to use these data to perform a test of $H_0$: $P_1=P_2=P_3$, where $pi=$the true proportion of spouse abusers like the ones in this study who would be arrested again within six months after receiving treatment$i.$

The following are the null and alternative hypotheses:

$H_0:p1=p2=p3$

$H_a:$At least one of $p_i$is different

Assume that all the conditions for performing the test in part (b) are met. The test yields $x^2=5.063$and a P-value of $0.0796$. We need to interpret this P-value in context.

$0.05$is the $p-$value. The null hypothesis is found to be false. Thus, there is adequate evidence to reject the assumption that all proportions are equal at the$5$percent significance level.