Sec. 5.5 x 2 WITH OVER ONE DEGREE OF FREEDOM 143 



pothesis used is correct. In fact P( x 2 — 7.07) = .03. If we have 

 decided in advance to reject a hypothesis when P < .05, H (pi = 

 p 2 = p 3 ) would be rejected, and we would say that the true fraction 

 of yes votes is not the same in all three districts. It is clear that after 

 such a decision it would not be valid to conduct a separate survey 

 in each district and then combine the evidence from these samples 

 on the assumption that we have three independent x 2 ' s testing the 

 same hypothesis, as is supposed in the theorem stated earlier in this 

 section. 



It appears from the samples given above that p x does equal p 2 , 

 but that ps is greater than p x or p 2 . This hypothesis could be tested 

 by the method just illustrated; but for the purposes of this discussion 

 districts 1 and 2 will be used to test the original conjecture that 

 p = .6 for the area covered by districts 1 and 2. 



For district 1, the expected number of yes votes is E(r) — .6(200) 

 = 120 votes. Therefore, x 2 = (15)7120 + (15) 2 /80 = 4.69 with 



1 D/F so that P = .030 by Table V. On the basis of this sample 

 evidence H (p = .6) is rejected at the 3 per cent level. 



For district 2, the same expected numbers are used because 200 

 votes were recorded in this sample also. Therefore, x 2 = (20) 2 /120 -f- 

 (20) 2 /80 = 8.33 with 1 D/F so that P = .003. This time H {p = .6) 

 is rejected more decisively. 



By the theorem of this section, x 2 = 4.69 + 8.33 = 13.02, with 



2 D/F so that P = .002 by Table V. Therefore H {p = .6) is re- 

 jected at the 0.2 per cent level upon the basis of the evidence in the 

 two 200-vote samples. 



The chi-square distribution with more than one degree of freedom 

 may be useful when the data are classified in a two-way table of r 

 rows and c columns. For example, a random sample of Republicans 

 and Democrats in a certain city might each be grouped on the basis 

 of three income brackets as follows: 



Annual Income 

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



Republican 200 50 8 258 



Democrat 120 20 3 143 



Sums 320 70 11 401 



This will be described as a 2 by 3 contingency table. Earlier in this 

 chapter 2 by 2 contingency tables were analyzed by means of the 

 chi-square distribution. 



