Sec. 5.4 TEST OF THE HYPOTHESIS H ( Pl = p 2 ) 137 



that among the 1000 flies sprayed 92.5 per cent were later classified 

 as dead. If these two sprays are equally toxic to the house flies, they 

 should tend to kill equally many flies per 500 sprayed. Therefore, 

 the probability of death can be taken as .925 on the general hypoth- 

 esis that the two sprays are equally toxic. This is equivalent to 

 the hypothesis, H (pi = Pz). Then the expected number dead out 

 of 500 in a cage is E{r) = .925(500) = 462.5. That leaves 37.5 as 

 the expected number of survivors for each spray since 500 flies were 

 sprayed with each spray. We then can extend formula 5.31 to obtain 

 the following: 



x 2 = (475 _ 462.5) 2 /462.5 + (450 - 462.5) 2 /462.5 



+ (25 - 37.5)737.5 + (50 - 37.5)737.5 = 9.01. 



This x 2 nas only one degree of freedom as before because there is 

 only one chance difference between the observed and expected num- 

 bers. Note that only one expected number need be calculated before 

 all the rest follow automatically from the border totals of the table. 

 Figure 5.31 and Table V clearly indicate that x 2 rarely would attain a 

 size of 9.01, or more, purely from sampling variations; therefore it is 

 concluded that the lethane spray is superior to the pyrethrum spray, 

 that is, the hypothesis that pi = p 2 is rejected, where p-i = true pro- 

 portion which would be killed by lethane and p 2 is the same for 

 pyrethrum over many trials. 



The technique just described also can be used to decide if two 

 random samples supposedly drawn from the same binomial popula- 

 tion actually are from a common population. For example, suppose 

 that two separate random samples were taken on the toxicity of a 

 lethane spray, with the following results: 



Dead Alive Sums 



Sample 1 480 20 500 



Sample 2 380 20 400 



Sums 860 40 900 



If p remained constant during this sampling it is best estimated as 

 p — 860/900 = .956 or 95.6 per cent. In the absence of any logical 

 predetermined hypothesis, the hypothesis H (pi = p 2 ) is tested, 

 where p x = true probability of death during the taking of the first 

 sample, and similarly for p 2 and the second sample. If the prob- 

 ability of death for any randomly designated fly stays fixed, Pi = p 2 . 



