basic statistis

Question

A total of 358 patients are tested for two diseases, Disease A and Disease B. Of these 358 patients, 42 have Disease A, 32 have Disease B, and 19 have both diseases. What is the χ2 statistic of your sample data? Write your answer correct to four decimal places.

Answer 1

|A ∩ B| = 19

 

|A ∩ B^c| = 42 - 19 = 23

 

|A^c ∩ B| = 32 - 19 = 13

 

 |A^c ∩ B^c| = 358 - 19 - 23 - 13 = 303

 

The table of true frequencies:

 

        | A      A^c    | Total

--------------------------

     B | 19    13    |   32

    B^c | 23   303   |  326

--------------------------

Total | 42   316   |  358

 

 

Use proportions to get the expected numbers.

 

A: 42/358 = 0.117318

B: 32/358 = 0.089385

 

Expected table:

 

        |       A               A^c                | Total
---------------------------------------------
     B |   3.754190     28.245810  | 32
    B^c | 38.245810   287.754190  | 326
---------------------------------------------
Total |       42             316              | 358

 

Now we can compute the test statistic  χ² = ∑ (O-E)²/E.

 

(19 - 3.754190)² / 3.754190 = 61.913415

 

(13 - 28.245810)² / 28.245810 = 8.228998

 

(23 - 38.245810)² / 38.245810 = 6.077391

 

(303 - 287.754190)² / 287.754190 = 0.807754

 

χ² = 61.913415 + 8.228998 + 6.077391 + 0.807754 = 77.027558

 

Rounded to 4 places this is χ² = 77.0276

 

By the way, we have df = 1, so the p-value is 1.685891 × 10^-18. Thus the evidence is overwhelming that having either disease increases your risk of getting the other one.