Question

A study conducted in Charlotte, North Carolina, tested the effectiveness of three police responses to spouse abuse: $\left(1\right)$advise and possibly separate the couple, $\left(2\right)$issue a citation to the offender, and $\left(3\right)$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 arrested within $6$months of the original incident.

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

b. State an appropriate pair of hypotheses for performing a chi-square test in this setting.

c. Assume that all the conditions for performing the test in part (b) are met. The test yields $x2=5.063x^2=5.063$and a$P-valueof0.0796$. Interpret this P-value.

d. What conclusion should we draw from the study?

Short answer

Part a. Random assignment helps to make unbiased result.

Part b. $H_0$: The true proportion of spouse abusers like the ones in the study who will be arrested within $6$months is the same for all three police responses.

$H_a$: The true proportion of spouse abusers like the ones in the study who will be arrested within $6$months is not same for all three police responses.

Part c. The p-value explains if all population probabilities are equal then the probability of obtaining a similar sample or more extreme is $7.96\%$.

Part d. There is not sufficient evidence to reject the claim of equal proportions in each group.

Step-by-step solution

Part a. Step 1. Given information

Given:

Part a. Step 2. Explanation

The conditions of testing a chi-square test are important to make as result accurate. From that conditions, the condition of random assignment is most important. Because, the random assignment helps to make groups of each treatment as similar as possible and to eliminate the effect of any variables that are not included in the study.

Part b. Step 1. Explanation

The null and alternative hypotheses:

$H_0$: The true proportion of spouse abusers like the ones in the study who will be arrested within $6$months is the same for all three police responses.

$H_a$: The true proportion of spouse abusers like the ones in the study who will be arrested within $6$months is not same for all three police responses.

Part c. Step 1. Explanation

The p-value explains if all population probabilities are equal then the probability of obtaining a similar sample or more extreme is$7.96\%$.

Part e. Step 1. Explanation

Decision: The $p-value>0.05$, fail to reject $H_0$.

Conclusion: There is not sufficient evidence to reject the claim of equal proportions in each group.