144 SAMPLING FROM BINOMIAL POPULATIONS Ch. 5 



It is likely that a person might wish to answer the following ques- 

 tion with the aid of the data in the above table. Is the announced 

 party affiliation of a voter in this city associated with that voter's 

 economic status? Or, in statistical terminology, the question can be 

 rephrased as follows: Let pi be the proportion of Republicans in this 

 city with incomes under $5000, p 2 be the same for Republicans with 

 incomes in the middle income group, p 3 the same for those Repub- 

 licans in the highest income bracket; and let pt' = the corresponding 

 proportions in the population for the Democrats in this city. The 

 subscript i takes the values 1,2, and 3. Then we wish to test the 

 more complex hypothesis: H (pi — p/, i = 1, 2, 3). As usual, 2?>i = 

 Sp t ' = 1 for i = 1, 2, and 3. 



The pi and p{ are unknown and will be estimated from the sample 

 observations on the assumption that H is correct. These estimates 

 of parameters will be obtained as before: p\ = pY = 320/401; p 2 = 

 pY = 70/401; and p 3 = pY = 11/401. It follows that the expected 

 number of Republicans in the lowest income stratum is (320/401) 

 (258) = 205.9. The other expected numbers are computed in a 

 similar manner and are shown in parentheses in the following table: 



Annual Income 

 Party Under $5000 $5000-$9999 $10,000 and Over Sums 



Republican 200(205.9) 50(45.0) 8(7.1) 258(258.0) 



Democrat 120(114.1) 20(25.0) 3(3.9) 143(143.0) 



Sums 320(320.0) 70(70.0) 11(11.0) 401(401.0) 



_ 2 (200 - 205.9) 2 (120 - 114.1) 2 , (3 - 3.9) 2 __ 



HenC6X = 20K9- " + ilLl— + --- + ~^9— = 2 - 35 - 



How many degrees of freedom does this sampling chi-square have? 

 In the process of estimating the pi and the p{ the expected numbers 

 in a column were required to add to the same sum as the observed 

 numbers for the same columns. This causes the deviations from 

 expectation in a column to be the negatives of each other. For exam- 

 ple, 200 - 205.9 = -(120- 114.1). Therefore, both these devia- 

 tions of observation from expectation cannot be chance occurrences. 

 There are, then, at most three chance deviations among the six which 

 go into the computation of the chi-square. Furthermore, the expected 

 numbers of Republicans in the three income classes must add to 258, 



