Chapter 10.1, Problem 10.30E

(a)

To determine

To Explain: the randomization important in this study.

Answer to Problem 10.30E

To exclude the effect of any lurking variables

Explanation of Solution

Given:

	No subsequent arrest	Subsequent arrest
Advise and separate	187	25
Citation	181	43
Arrest	175	39

Assume:

  $\alpha =0.05=5\%$

Randomization are very important such that the results of being arrested or not is unaffected by any lurking variables. A lurking variable is a variable that has a very important effect on the relationship among the variables in a study but it is not one of the explanatory variables studied.

(b)

To determine

To Explain: that there is sufficient evidence from this study to conclude that the treatment imposed on an abuser is related to the likelihood of the abuser’s subsequent.

Answer to Problem 10.30E

There is no enough evidence to help the claim that the treatment imposed on an abuser is related to the likelihood of arrest.

Explanation of Solution

Given:

	No subsequent arrest	Subsequent arrest
Advise and separate	187	25
Citation	181	43
Arrest	175	39

Formula used:

  $\begin{array}{c}{\chi }^2={\displaystyle \sum \frac{{\left(O-E\right)}^2}{E}}\\ E=np\end{array}$

Calculation:

The null hypothesis statement is that there is no connection between the variables, where the alternative hypothesis statement is that there is a connection between the variable.

  $H_0:$ There is no connection between Treatment and Arrest.

  $H_1:$ There is a connection between Treatment and Arrest.

The row and column totals are

	No subsequent arrest	Subsequent arrest	Total
Advise and separate	187	25	212
Citation	181	43	224
Arrest	175	39	214
Total	543	107	650

  $\begin{array}{c}{E}_{11}=\frac{r_1\times c_1}{n}=\frac{212\times 543}{650}=177.1015\\ E_{12}=\frac{r_1\times c_2}{n}=\frac{212\times 107}{650}=34.8985\\ E_{21}=\frac{r_2\times c_1}{n}=\frac{224\times 543}{650}=187.1262\\ E_{22}=\frac{r_2\times c_2}{n}=\frac{224\times 107}{650}=36.8738\\ E_{31}=\frac{r_3\times c_1}{n}=\frac{214\times 543}{650}=178.7723\\ E_{32}=\frac{r_3\times c_2}{n}=\frac{214\times 107}{650}=35.2277\end{array}$

The value of the test- statistic is then

  $\begin{array}{c}{\chi }^2={\displaystyle \sum \frac{{\left(O-E\right)}^2}{E}}\\ =\frac{{\left(187-177.1015\right)}^2}{177.1015}+\frac{{\left(25-34.8985\right)}^2}{34.8985}\\ +\frac{{\left(181-187.1262\right)}^2}{187.1262}+\frac{{\left(43-36.8738\right)}^2}{36.8738}\\ +\frac{{\left(175-178.7723\right)}^2}{178.7723}+\frac{{\left(39-35.2277\right)}^2}{35.2277}\\ =5.0627\end{array}$

The degree of freedom is

  $df=(r-1)\left(c-1\right)=\left(3-1\right)\left(2-1\right)=2$

The P-Value is the probability of getting the value of the test statistic, or a value more extreme or $df=2:$

  $0.05<P<0.10$

If the P-value is equal or less than to the significance level, then the alternative hypothesis is accepted and null hypothesis is rejected:

  $P>0.05\Rightarrow {\text{Fail to reject H}}_0$

There is no enough evidence to help the claim that the treatment imposed on an abuser is related to the likelihood of arrest.