Elementary Biometrical Inferences 



533 



(number of rows — 1) times (number of 

 columns — 1) in a contingency table. 



Suppose three samples were scored four 

 alternative ways to give the results shown 

 in Figure A-5A. The procedure followed is 

 the same as that already described for the 

 2 X 2 or four-fold table (note that Yates' 

 correction is not applicable in any larger 

 table). There are 6 degrees of freedom. 

 If one tests at the 5% level, Figure A-3 

 shows that x%) has to be greater than 12.5 

 if one is to reject the null hypothesis, 

 namely, that all the samples and types can 

 be represented by the same parameters. 

 Moreover, finding that x 2 is less than 1.6 

 would mean that the same parameters 

 would produce samples varying this little 

 from the ideally expected values only 5% 

 of the time. In that case one would reject 

 the samples as being random, suspecting 

 that there was some hidden bias in the 

 collection and/or the scoring of the data. 

 The decision that neither obtains can be 

 seen from Figure A-5D. Consequently, 

 one accepts the null hypothesis that these 

 samples are not statistically different at the 

 5% level of significance. 



Assume, however, that chi-square had 

 been 14.1 in the preceding example. One 

 would reject the null hypothesis at the 5% 

 level but could accept it at the 1% level 

 of significance (meaning that these samples 

 have more than 1%, but less than 5%, 

 chance of having the same parameters). 

 Assuming that such a result was obtained 

 in an unbiased manner it might be due to 

 the fact that (a) the null hypothesis is true 

 but one happened to collect data (as will 

 happen by chance one time in 20) which 

 varied at least this much from those ex- 

 pected, or (b) the null hypothesis is in- 

 correct. Even if the hypothesis at the 5% 

 level is rejected, one may wish to test the 

 data further, using smaller contingency 

 tables to determine which samples or out- 

 comes are consistent or inconsistent with 



each other according to a null hypothesis. 

 Note here that the observed values in 

 a contingency table furnishing the larg- 

 est contributions to chi-square are those 

 most responsible for the rejection of the 

 hypothesis. 



PROBLEMS 



A.28. 



A. 29. 



A cross yields 20 offspring of one 

 type and 40 of another. A month 

 later the same cross produces 15 of 

 the first type and 15 of the second. 

 Do these results differ significantly? 

 Ten sets of identical twins are se- 

 lected ; only one member (the same 

 one) of each pair is given a particular 

 drug daily for 10 days. All indi- 

 viduals are weighed before and after 

 this period. The changes to the 

 nearest whole pound are as follows: 



Twin 



Analyze the results of this experi- 

 ment statistically. 

 A. 30. Among the women of population A 

 are 10 blondes, 5 redheads, and 15 

 of other hair color. In population 

 B there are 7, 7, and 6, respectively; 

 whereas in population C the tally is 

 8, 4, 8, respectively. Are these 

 populations the same with respect 

 to the relative frequency of these 

 hair color types? 



