Chapter 10.1, Problem 10.24E

To determine

To Explain: the relationship between treatment and success statistically significant and carry out significance test and conclude.

Answer to Problem 10.24E

  ${\chi }^2=10.5$

  $P<0.5\Rightarrow {\text{Reject H}}_0$

There is no enough evidence to help the claim that there is a connection between Treatment and Successes.

Explanation of Solution

Formula used:

  ${\chi }^2={\displaystyle \sum \frac{{\left(O-E\right)}^2}{E}}$

Calculation:

Suppose

  $\alpha =0.05=5\%$

Number of the subjects in the treatment will be the column totals

	Success	Failure	Total
Desipramine	14	10	24
Lithium	6	18	24
Placebo	4	20	24
Total	24	48	72

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

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

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

The expected frequencies $E$

  $\begin{array}{c}{E}_{11}=\frac{r_1\times c_1}{n}=\frac{24\times 24}{72}=8\\ E_{12}=\frac{r_1\times c_2}{n}=\frac{24\times 48}{72}=16\\ E_{21}=\frac{r_2\times c_1}{n}=\frac{24\times 24}{72}=8\\ E_{22}=\frac{r_2\times c_2}{n}=\frac{24\times 48}{72}=16\\ E_{31}=\frac{r_3\times c_1}{n}=\frac{24\times 24}{72}=8\\ E_{32}=\frac{r_3\times c_2}{n}=\frac{24\times 48}{72}=16\end{array}$

The value of the test- statistic is

  $\begin{array}{c}{\chi }^2={\displaystyle \sum \frac{{\left(O-E\right)}^2}{E}}\\ =\frac{{\left(14-8\right)}^2}{8}+\frac{{\left(10-16\right)}^2}{16}\\ +\frac{{\left(6-8\right)}^2}{8}+\frac{{\left(18-16\right)}^2}{16}\\ +\frac{{\left(4-8\right)}^2}{8}+\frac{{\left(20-16\right)}^2}{16}\\ =10.5\end{array}$

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.

  $0.01<P<0.001$

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

  $P<0.05\Rightarrow {\text{Reject H}}_0$

There is no enough evidence to help the claim that there is a connection between Treatment and Successes.