Question

Two allergists recorded the main area of allergy for new patients during a month.
The data is shown below:

Allergies

Pollen

Food

Mold

Pets

Doctor 1

8

5

9

8

Doctor 2

34

6

11

23

Using   = .05, test the claim that the allergy diagnosis and the doctor that treated the patients are independent.

Group of answer choices

There is not evidence to reject the claim that the allergy diagnosis and the doctor are not related because the test value 6.125 < 7.815

There is not evidence to reject the claim that the allergy diagnosis and the doctor are not related because the test value 2.074 > 15.507

There is evidence to reject the claim that the allergy diagnosis and the doctor are not related because the test value 7.815 > 6.125

There is evidence to reject the claim that the allergy diagnosis and the doctor are not related because the test value 15.507 > 2.074

Answer 1

For each of the 8 cells above, we obtain the expected frequencies as:
E_i = (Sum of column i)*(Sum of row i) / Grand Total

Also we obtain the chi square test statistic contribution for each cell as:

The value in the circular bracket is the expected value for that cell while that in the square bracket is the chi square test statistic contribution.

Therefore the chi square test statistic for the whole test is obtained here as:

Now the degrees of freedom here is computed as:
Df = (num of rows - 1)(num of columns - 1) = 3

Therefore for 3 degrees of freedom, and for 0.05 level of significance, we have from the chisquare distribution tables:

Therefore 7.815 is the required critical value for the test here.

As the chi square test statistic = 6.125 < 7.815, therefore the test is not significant and we dont have enough evidence here that the two variables are associated. Therefore A is the correct answer here.