Question

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.

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

Short answer

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.

Step-by-step solution

Given Information

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

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

Hypotheses test

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:

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

$=125.7399$

Graph

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

Chi-Square Distribution

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

Hypothesis rejection

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.