142 SAMPLING FROM BINOMIAL POPULATIONS Ch. 5 



ideas, suppose that some respected group of persons has conjectured 

 that 60 per cent of the voters in a certain area will vote yes on a 

 given economic question, and that this conjecture is to be tested by- 

 means of three samples taken in the three districts in the area in- 

 volved. It will be assumed that these districts contain equally many 

 voters. 



Before the hypothesis that p = .60 for the whole area is tested, 

 it is of interest to determine if the three districts are the same bi- 

 nomial population with respect to yes and no votes on the economic 

 question which is to be asked the voters. Hence it is supposed that 

 a poll of 200 randomly chosen voters in each district gave these 

 results : 



Number Voting 

 District Yes No Sum 



If the whole of each district has the same fraction, p, of yes votes, 

 the best estimate of p is p = 330/600 = .55 or 55 per cent. If this is 

 used as the probability that a randomly chosen voter in a given 

 district will vote yes, the expected number of yes votes in each dis- 

 trict is .55(200) = 110. That leaves 90 as the expected number of 

 noes; hence 



, (105 - 110^ 2 (75 - 90) 2 



x 2 = + ••• + = 7.07. 



110 90 



It is seen that the observed number of yeses in the first district is 5 

 below expectation; hence the number of noes is 5 above expectation, 

 and only one chance difference between observation and theoretical 

 expectation exists. The same can be said for district 2; but since we 

 know that the yes vote in district 2 was 10 below expectation and 

 that in district 1 is 5 below expectation, it follows that the number 

 of yeses from district 3 must have been 15 above expectation. Hence 

 only two chance deviations are involved basically, and this x 2 has 2 

 degrees of freedom. 



The specific hypothesis being tested is that the true proportions 

 of yes votes in the three districts are equal. Table V indicates that 

 it is rather uncommon during sampling experience for a x 2 with 2D/F 

 to become as large as the 7.07 observed for these samples if the hy- 



