The table provides a recent survey of the youngest online entrepreneurs whose net worth is estimated at one million dollars or more. Their ages range from $17$to $30$. Each cell in the table illustrates the number of entrepreneurs who correspond to the specific age group and their net worth. Are the ages and net worth independent? Perform a test of independence at the $5\%$significance level.

$\begin{array}{|lllll|}\hline \text{Age Groupl Net Worth Value (in millions of US dollars)}& \mathbf{1}\mathbf{-}\mathbf{5}& \mathbf{6}\mathbf{-}\mathbf{24}& \ge 25& \text{Row Total}\\ 17-25& 8& 7& 5& 20\\ 26-30& 6& 5& 9& 20\\ \text{Column Total}& 14& 12& 14& 40\\ \hline\end{array}$

The null hypothesis is shown below:

$H_0$: The ages and net worth is independent.

Against the alternative hypothesis as shown below:

$H_a:$The ages and net worth is dependent.

The degrees of freedom can be calculated by the formula given below:

$df=(numberofcolumns-1)(numberofrows-1)$

Therefore,

$df=(numberofcolumns-1)(numberofrows-1)$

$=(3-1)(2-1)$

$=2\times 1$

$=2$

From above calculation, it is clear that the distribution for the test is ${\chi }_2^2$.

The observed value table is already given in the textbook. Calculate the expected frequencies by using the formula shown below:

$E=\frac{\left(\text{row total}\right)\left(\text{column total}\right)}{\text{overall total}}$

All calculations can be done in an excel worksheet. Hence, the expected (E) values table is shown below:

The test statistic of independence test is given below:

$Teststatistic=\sum _{(i\times j)}\frac{(O-E{)}^2}{E}$

To calculate $\frac{(O-E{)}^2}{E}$apply formula $=\left((B4-B11{)}^2\right)/B11$in cell $B17$and drag the same formula up to cell $D18$. After that, take the total of columns total and rows total. The table of test statistic is shown below:

Hence, the test statistic is $1.76$.

The $p$-value can be calculated in excel by using CHIDIST ( ) formula as shown below:

Hence, the$p$value is$0.41$

The Chi-square sketch is given below

Alpha:$0.05$

Decision: Do not reject the null hypothesis $H_0$

Reason for decision: Because $p$-value $>\alpha$

Conclusion: There is sufficient evidence to insure that the ages and net worth is independent.