132 SAMPLING FROM BINOMIAL POPULATIONS Ch. 5 



population parameters for which the best regions of rejection can 

 be defined. The best region of rejection, among several choices, is 

 the one which for a given a will make ft the smallest; that is, for a 

 fixed probability of rejecting a true hypothesis it will give the lowest 

 probability of accepting a false hypothesis in consideration of the 

 other possible hypotheses. 



Statistical research has shown that a good function to use in the 

 solution of the problem set up for this section is one which is called 

 chi-square, and is denoted by the symbol x 2 . Its magnitude depends 

 upon the numbers of individuals, or other units, observed in the 

 sample to fall into each of the possible classes of attributes. It also 

 depends upon the numbers which are expected mathematically to 

 fall in those classes, which in turn depends upon the predetermined 

 hypothesis regarding the population parameter p. For example, sup- 

 pose that we have sufficient reason to believe that one-half the off- 

 spring of guinea pigs should be males. The predetermined hypothesis 

 now is that p = 1/2. If a sample group of progeny selected at random 

 from a whole population of actual or possible progeny is found to 

 have 38 males and 32 females, is it reasonable to believe that this is 

 a sample from a population for which p = 1/2? The mathematically 

 expected number of males out of 70 offspring is E (r) = (1/2) (70) = 

 35; hence the number of males in the sample is 3 greater than ex- 

 pectation. It follows automatically that there are 3 fewer females 

 than expected mathematically. 



The function x 2 will be defined by the following formula: 



v ^ (observed number in class — expected number) 2 

 expected number in class 



where the summation includes two terms, one for males and one for 

 females. It is apparent from this formula that if the observed num- 

 bers in the two classes agree well with those numbers which are 

 expected mathematically considering the assumed magnitude of p, 

 X 2 will be relatively small; but if the numbers observed to fall in 

 each class notably disagree with those expected from the predeter- 

 mined hypothesis, x 2 will be relatively large. The decision that x 2 

 is relatively large or small is based upon the proportion of all such 

 sample values of x 2 which would be at least that large if the hypoth- 

 esis being tested were, in fact, true. 



For the illustration above, x 2 = (38 - 35) 2 /35 + (32 - 35) 2 /35 

 = 0.51. The remaining question is: Is it reasonable to suppose that 



