Chi-Squared Test for Independence

Date: 05/24/2000 at 03:34:00
From: Leanna
Subject: Probability

Saint Botulism's Hospital commissioned an independent study to 
determine if there was a difference between the quality of care 
provided by hospital-trained nurses and tertiary-trained nurses. Two 
hundred nurses were rated as either providing Superior Service (S) or 
Adequate Service (A). The results are as follows:

                          Superior (S)     Adequate (A)
     Hospital-trained (H)     48                12
     Tertiary-trained (T)     76                64

Are the events H and S independent? Justify your answer 
mathematically.

Date: 05/24/2000 at 11:58:06
From: Doctor Anthony
Subject: Re: Probability

Do a chi-squared test using a contingency table:

     Observed
     ---------

              | S        A |  Total
        ------|------------|---------
         (H)  | 48      12 |   60
         (T)  | 76      64 |  140  
        ------|------------|---------
        Total |124      76 |  200

     Expected
     ---------

              | S        A |  Total
        ------|------------|---------
         (H)  | 37.2   22.8|   60
         (T)  | 86.8   53.2|  140  
        ------|------------|---------
        Total |124      76 |  200

The entries in the second table are calculated as follows:

     37.2 = 124 * 60/200, 22.8 = 76 * 60/200, and so on.

To calculate chi-squared we make up a third table. Since there is only 
1 degree of freedom we use Yates's correction:

       O    E     |O-E|   Y = |O-E|-1/2   Y^2/E
     -------------------------------------------
      48   37.2   10.8        10.3        2.852
      12   22.8   10.8        10.3        4.653
      76   86.8   10.8        10.3        1.222
      64   53.2   10.8        10.3        1.994
                                        --------
                                   X^2 = 10.72

We compare this with the tabular value X^2 (5%) (V = 1) = 3.84

So our value of 10.72 is very significant at the 5% level and we 
conclude that H and S are NOT independent.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/