According to Dan Lenard, an independent insurance agent in the Buffalo, N.Y. area, the following is a breakdown of the amount of life insurance purchased by males in the following age groups. He is interested in whether the age of the male and the amount of life insurance purchased are independent events. Conduct a test for independence.

$\begin{array}{|llllll|}\hline \text{Age of Males}& \text{None}& <\mathbf{\backslash (}\mathbf{200}\mathbf{,}\mathbf{000}& \mathbf{\backslash )}\mathbf{200}\mathbf{,}\mathbf{000}\mathbf{-}\mathbf{\backslash (}\mathbf{400}\mathbf{,}\mathbf{000}& \mathbf{\backslash )}\mathbf{401}\mathbf{,}\mathbf{001}\mathbf{-}\mathbf{\backslash (}\mathbf{1}\mathbf{,}\mathbf{000}\mathbf{,}\mathbf{000}& \mathbf{\backslash )}\mathbf{1}\mathbf{,}\mathbf{000}\mathbf{,}\mathbf{001}\mathbf{+}\\ 20-29& 40& 15& 40& 0& 5\\ 30-39& 35& 5& 20& 20& 10\\ 40-49& 20& 0& 30& 15& 30\\ 50+& 40& 30& 15& 0& 10\\ \hline\end{array}$

The table with observed values

$\begin{array}{|ccccccc|}\hline \text{Age}& \text{None}& <\$200\mathrm{k}& \$200\mathrm{k}-\$400\mathrm{k}& \$401\mathrm{k}-\$1,000\mathrm{k}& \$1,000,001+& \text{Total}\\ 20-29& 40& 15& 40& 0& 5& 100\\ 30-39& 35& 5& 20& 20& 10& 90\\ 40-49& 20& 0& 30& 0& 30& 80\\ 50+& 40& 30& 15& 15& 10& 110\\ \text{Total}& 135& 50& 105& 35& 55& 380\\ \hline\end{array}$

The table with expected values

$\begin{array}{|ccccccc|}\hline \text{Age}& \text{None}& <\$200\mathrm{k}& \$200\mathrm{k}-\$400\mathrm{k}& \$401\mathrm{k}-\$1,000\mathrm{k}& \$1,000,001+& \text{Total}\\ 20-29& 35.526316& 13.157895& 27.631579& 9.210526& 14.473684& 100\\ 30-39& 31.973684& 11.842105& 24.868421& 8.289474& 13.026316& 90\\ 40-49& 28.421053& 10.526316& 22.105263& 7.368421& 11.578947& 80\\ 50+& 39.07895& 14.47368& 30.39474& 10.13158& 15.92105& 110\\ \text{Total}& 135& 50& 105& 35& 55& 380\\ \hline\end{array}$

We want to test following hypotheses

$H_0$: The age of the male and the amount of life insurance purchased are independent events

$H_1$: The age of the male and the amount of life insurance purchased are not independent events

Since there are $4$age groups and $5$salary groups, the number of degrees of freedom is

$(4-1)·(5-1)=12$

We are using ${\chi }^2$distribution. Our test statistic is given by:

${\chi }^2=\sum _{i=1}^4\sum _{j=1}^5\frac{{\left(O{b}_{i,j}-E{x}_{i,j}\right)}^2}{E{x}_{i,j}}$

$=125.7399$

Using the applet for ${\chi }^2$distribution, we can see that $p$-value is $0$:

Chi-Square Distribution

$X~{\chi }_{\left(df\right)}^2$

Taking $\alpha =0.05$, we can see that $p<\alpha$. This means that we reject the null hypothesis.

There is evidence to conclude that the age of the male and the amount of life insurance purchased are not independent events.