Question

Suppose that $600$thirty-year-olds were surveyed to determine whether or not there is a relationship between the level of education an individual has and salary. Conduct a test of independence

$\begin{array}{|lllll|}\hline \begin{array}{l}\text{Annual}\\ \text{Salary}\end{array}& \begin{array}{l}\text{Not a high school}\\ \text{graduate}\end{array}& \begin{array}{l}\text{High school}\\ \text{graduate}\end{array}& \begin{array}{l}\text{College}\\ \text{graduate}\end{array}& \begin{array}{l}\text{Masters or}\\ \text{doctorate}\end{array}\\ <\backslash (30,000& 15& 25& 10& 5\\ \backslash )30,000-\backslash (40,000& 20& 40& 70& 30\\ \backslash )40,000-\backslash (50,000& 10& 20& 40& 55\\ \backslash )50,000-\backslash (60,000& 5& 10& 20& 60\\ \backslash )60,000+& 0& 5& 10& 150\\ \hline\end{array}$

Short answer

Decision: Reject the null hypothesis $H_0$

Conclusion: There is sufficient evidence to ensure that the salary is dependent on the level of education an individual has.

Step-by-step solution

Given Information

The null hypothesis is shown below:

$H_0$: The salary is independent of the level of education an individual has.

Against the alternative hypothesis as shown below:

$H_a$: The salary is dependent of the level of education an individual has.

Degrees of freedom

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

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

Therefore,

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

$=(4-1)(5-1)$

$=3\times 4$

$=12$

From the above calculation, it is clear that the distribution for the test is ${\chi }_{12}^2$.

Tables

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

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

All calculations can be done in excel worksheet. Hence, the expected$\left(E\right)$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 $=\left((B4-B14{)}^2\right)/B14$in cell B21 and drag the same formula up to cell F24. After that, take total of columns total and rows total. The table of the test statistic is shown below:

Hence, the test statistic is $255.77$.

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

Hence, the$p$- value is$0.000$

Graph

Chi-square sketch is given below:

Decision, Reason and conclusion

Alpha:$0.05$

Decision: Reject the null hypothesis $H_0$

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

Conclusion: There is sufficient evidence to ensure that the salary is dependent of the level of education an individual has.