91Ó°ÊÓ

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 ${\mathrm{χ}}^2$distribution. Our test statistic is given by:

${\mathrm{χ}}^2=\underset{i=1}{\overset{4}{∑}}\underset{j=1}{\overset{5}{∑}}\frac{{\left(O{b}_{i,j}-E{x}_{i,j}\right)}^2}{E{x}_{i,j}}$

$=125.7399$

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

Chi-Square Distribution

$X~{\mathrm{χ}}_{\left(df\right)}^2$

Taking $\mathrm{Î\pm }=0.05$, we can see that $p<\mathrm{Î\pm }$. 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.