Question

A research firm was interested in knowing whether income level and location of service for procedures were independent. The following data were provided.

	Clinic	Hospital	Private
<$30,000	85	16	6
$30,000 to $49,999	102	27	13
$50,000 to $99,999	36	22	15
> $99,999	15	23	25

Use a chi-square test of independence to determine whether income level and service location are independent. Test the null hypothesis the service location is independent of income at α = 0.01.

Answer 1

As we are trying to test here whether the 2 variables are independent, we first compute the grand total of each row and column as:

Now the expected value for each of the 12 cells is computed here as:
E_i = (Row_i Total)(Column_i Total) / Grand Total

The expected value for each cell thus is computed here as:

The expected values are given in the circular bracket above.

Also now the chi square test statistic contribution for each cell here is computed as:

This is given in the square bracket for each cell.

The chi square test statistic now is computed here as:

Now, degrees of freedom here is computed as:
DF = (num of rows - 1)(num of columns - 1) = (4 - 1)(3 - 1) = 6

Now the p-value here is computed as:

As the p-value here is less than 0.01, therefore the test is significant and we can reject the null hypothesis here and conclude that we have sufficient evidence that the 2 variables are not independent here.