Sec. 5.5 x 2 WITH OVER ONE DEGREE OF FREEDOM 141 



theoretical numbers 900, 300, 300, and 100, respectively, to justify 

 the rejection of H ? In these circumstances 



2 (885 - 900) 2 (310 - 300) 2 (292 - 300) 2 (113 - 100) 2 



900 300 300 100 

 = 2.49, 



by analogy with previous problems when x 2 had only one degree of 

 freedom. How many degrees of freedom does this x 2 have, that is, 

 how many of the deviations from theoretical expectations can be 

 considered due to chance? The observed numbers and the expected 

 numbers each must add to 1600, hence there cannot be more than 

 3 degrees of freedom. As this is the only such restriction, there are 

 just 3 degrees of freedom. Naturally a sampling chi-square which 

 results from 3 chance deviations usually will be larger than one based 

 on fewer degrees of freedom because more "room" must be left for 

 sampling fluctuations. It is seen in Table V that a x 2 with 3 degrees 

 of freedom will exceed the observed value, 2.49, about one-third of 

 the time when H is correct; that is, P = .33. Therefore, the hy- 

 pothesis # (9 ARh+ : 3 ARh- : 3 ORh+ : 1 ORh-) is quite ac- 

 ceptable in view of this sample evidence. 



Another circumstance which produces a x 2 with more than one 

 degree of freedom is encountered when the same hypothesis is tested 

 more than once by means of successive but independent random 

 samples believed to have been taken under the same conditions. It 

 may not be reasonably possible to obtain a convincingly large sample 

 during any one experiment or survey so that some means of ac- 

 cumulating statistical evidence from two or more studies is needed. 

 This problem can be solved with the aid of the following theorem. 



Theorem. If two or more sample chi-squares are obtained from 

 independent random samples, the sum of these chi-squares fol- 

 lows the chi-square distribution for a number of degrees of 

 freedom equal to the sum of those for the chi-squares so added. 



Obviously, the process to which the above theorem refers would 

 make no practical sense unless each x 2 were obtained while the same 

 hypothesis was being tested. It also is important to be assured that 

 all the samples have been drawn from the same binomial population, 

 regardless of the truth of the hypothesis H because nothing is ac- 

 complished by such a study if several different populations are in- 

 volved. We are trying to test one predetermined hypothesis which 

 supposedly applies to a fixed set of conditions. To illustrate these 



