Sometimes we compare samples of different populations for 
   certain characteristics.  This data is often presented in a two 
   way table.  The test we use is a chi-square test of homogeneity.  
   This is one of two similar chi-square tests, the other being a 
   test for independence.
                     Relapse    Relapse   Total
                       No         Yes
     Desipramine 14            10         24
     Lithium       6           18         24
     Placebo       4           20         24
     Total         24          48         72
    The data represent random samples of cocaine addicts treated with 
    different drugs to reduce the likelyhood of a relapse.  Are the 
    proportion of addicts who relapse the same for each category?
      Step 1:    H: The proportion of cocaine addicts who relapse  
                   0
                 is the same for the different treatments. 
     H: The proportion of cocaine addicts who relapse  is not the 
       a
     same for all treatments. 
      Step 2:     Assumptions:  Our data are counts. 
      We have random samples of the populations, as stated.
                                                                                       c2
       To find expected counts we enter the data in [A], run the     test, 
       and check [B].
                          é  14 10 ù
          o = A =ê                        ú
         [  ]    [   ]    ê   6     18 ú
                          ê   4     20 ú                             é             ù
                          ë               û                          ê   8 16 ú
                                                       e = B =
                                                     [  ]    [   ]   ê   8 16 ú
                                                                     ê   8 16 ú
                                                                     ë             û
        All expected counts are more than 5.
                           (o- e)2
       Step 3:     c2 =å            =10.5
                               e
              Degrees of freedom are the number of rows 
              minus 1 times the number of columns minus 1.  
                    df =(r- 1)(c- 1)=(3- 1)(2- 1)=2