Elementary Biometrical Inferences 



529 



The chi-square test can be used to deter- 

 mine whether a sample is consistent with 

 an hypothesized 9:3:3: 1 ratio, for ex- 

 ample. If a 9:3:3:1 ratio were being 

 tested, the ideally expected numbers in a 

 group of 80 individuals would be 45, 15, 

 15, and 5, respectively. Since there are 

 four classes, there are three variables or 

 degrees of freedom. If we observed 40, 20, 

 12, and 8, respectively, we would calculate 



2 (40-45) 2 (20 - 15) 2 



X(3) — " Ar 



45 



15 



(12 - 15) 2 (8 - 5) 2 

 + " IT - + — 5- " = 4 ' 6 * 



(The term 3^» the Yates' correction, is 

 not applicable if there is more than one 

 degree of freedom.) Since the probability 

 lies between 0.20 and 0.25, the difference is 

 nonsignificant, and one accepts the null 

 hypothesis. 



It is interesting to note that the prob- 

 ability of obtaining a x value equal to or 

 greater than 0.004 for one degree of free- 

 dom, 0.1 for two degrees of freedom, etc. 

 is 0.95. It follows the probability of ob- 

 taining x 2 values smaller than these must 

 be 0.05. Such low values in an actual test 

 indicate that the agreement between ob- 

 servation and expectation is suggestively 

 better than expected. The question of 

 whether the data represent authentic ran- 

 dom samples may be legitimately raised in 

 such cases. 



PROBLEMS 



A. 20. A person with woolly hair marries 

 a nonwoolly-haired individual ; they 

 have 8 children, 7 woolly-haired and 

 1 nonwoolly-haired. Test the hy- 

 pothesis that woolly hair is due to a 

 rare, completely dominant gene. 



A.21. Given the data in A. 20, test the hy- 

 pothesis that woolly hair is due to a 

 completely recessive mutant. 



A. 22. A penny is tossed seven times. One 

 time it falls on edge, five times it 

 falls heads, and once it falls tails. Is 

 this an "honest" coin? 



A. 23. A test cross produces 57 individuals 

 of A phenotype and 43 of A' pheno- 

 type. Is one pair of genes involved ? 



A. 24. Given the data in A. 23, test the 

 hypothesis that one parent is a 

 dihybrid and that the A phenotype 

 is obtained only when two particular 

 nonalleles are present. 



A.25. In a sample of 540, X = 90. What 

 is the value of chi-square if you 

 hypothesize that p = 34? Do you 

 accept this hypothesis? 



A. 26. Among 60 individuals the pheno- 

 types are 8 A, 12 B, 20 C, and 20 D. 

 Test the hypothesis that: 



(a) A B C D are in the relative 

 proportion 1:3:3:9. 



(b) All four phenotypes have an 

 equal chance of occurring. 



(c) The ideal ratio is 1A : 3B : 5C : 

 7D. 



A. 27. A random sample from a natural 

 population contains 65 AA, 95 Aa, 

 and 40 aa individuals. Test the 

 hypothesis (after consulting Chapter 

 15) that: 



(a) The frequency of a in the 

 population gene pool is 0.5. 



(b) This sample is consistent with 

 the population being in genetic 

 equilibrium for this locus, if 

 you assume that the observed 

 gene frequency for a is also the 

 population frequency. 



E. Comparisons Between Statistics 

 (Figure A-1E) 



1. Involving One Variable 



a. Observed difference vs. expected standard 

 deviation. Suppose that a sample (A) pro- 

 vided 20 males and 30 females, whereas a 



