138 SAMPLING FROM BINOMIAL POPULATIONS Ch. 5 



As usual, the expected number killed during the first sampling is 

 computed to be E (r) = .956(500) = 478, which deviates from the 

 observed number by only 480 — 478 = +2. It follows that the other 

 observed numbers also differ from their expected numbers by 2 in 

 one direction or the other. Hence 



„ (+2) 2 (-2) 2 (-2) 2 (+2) 2 



x 2 = ^r- + - — ~ + - — - + ^- = °- 422 > 



478 382 22 18 



with 1 D/F. It is learned from Table V that P = .52 ; hence H is 

 accepted readily, and it is considered that the two samples were 

 taken under conditions which kept the probability of death constant. 

 It is not always true that the population can be kept the same under 

 repeated sampling; hence it is well to check this matter before dif- 

 ferent conditions (such as use of different insecticides) are purposely 

 introduced so that their effects can be studied. 



Problem 5.41. Suppose that two sample polls of votes for two candidates 

 for a public office are taken, one from among residents of cities with at least 

 25.000 population, the other from among residents not in any incorporated 

 town or city. If the results were as given below would you accept the statement 

 that place of residence was unrelated to voting preference in this election? If 

 so, the two samples are from a common binomial population. 



Votes for 

 A B Sums 



Rural 620 380 1000 



Urban 550 450 1000 



Sums 1170 830 2000 



Over both the rural and urban samples 58.5 per cent voted for A. 

 If both samples are from the same binomial population, p = .585 is 

 the best available estimate of p, the true fraction who favor A. Hence 

 the hypothesis H (p r = p u ) will be tested by means of the x 2 distri- 

 bution. The expected number of rural residents out of 1000 who favor 

 A is .585(1000) = 585. It deviates from the observed number by 

 620 - 585 = +35; hence 



X 2 = (35) ; 



1 1 1 1 



4- 4- 4- 



585 585 415 415. 



= 10.09, 1 D/F. 



It is apparent from Table V that H should be rejected because 

 P = .002. It is concluded that p r actually is > p«; that is, the resi- 



