520 



APPENDIX 



1). Comparing Ouskkvkd wim Kxpkcted 

 Statistics I Figure A-1D) 



I. The Binomial Test of a Parameter 

 Involving One Variable 



From a certain cross, genetic theory pre- 

 dicts a 1 : 1 ratio (p = 0.50) in F,. Among 

 6 individuals one expects, according to the 

 binomial expansion, to observe 3 of one type 

 and 5 of another '}{§ of the time. This is 

 the outcome most frequently obtained, all 

 others occurring with lower frequency. Sup- 

 pose, however, that one actually observes 

 that all 6 are of one type. Must one con- 

 sider this observation of no statistical sig- 

 nificance and due only to chance variation? 

 Or, is the difference statistically significant, 

 indicating that expectation and observation 

 do not always agree? This question can 

 be answered by considering the probability 

 of obtaining all 6 alike on the basis of our 

 hypothesis. According to this expectation, 

 the probability that a single individual will 

 be of the first type is }/$, and the probability 

 that it will be of the second type is also %. 

 The probability that all 6 will be of the 

 first type is (J/£) 6 ; the probability that all 6 

 will be of the second type is also (J^) 6 . 

 And the probability that either all 6 will be 

 of the first type or all of the second is 

 ( l A) 6 + (3^) 6 = 0.03. But since the prob- 

 ability of this outcome, if the hypothesis 

 holds true, is so low, one must conclude 

 one of two things. Either the hypothesis 

 is correct but a very improbable situation 

 has occurred, or else the hypothesis does 

 not fit the observations. Since an event 

 with a probability of 0.03 is expected to 

 occur only 3 times in a hundred trials, the 

 latter alternative is chosen. It is con- 

 cluded, therefore, that the hypothesis is 

 probably incorrect. 



In general, to test whether an observed 

 result is consistent with a parameter, one 



tests the null hypothesis, thai is, the likeli- 

 hood thai the statistic really has the hypo- 

 thesized parameter. Accordingly, one cal- 

 culates the total probability with which he 

 would expert to obtain from the parameter 

 a statistic which is as extreme as, or more 

 extreme than, the observed statistic. If 

 this probability is low (by convention, 0.05 

 or less), it can be concluded that observa- 

 tion and expectation do not agree. One 

 rejects the hypothesis with 95% confidence 

 and at a 5% level of significance (5% chance 

 of rejecting the hypothesis when it is really 

 true). If the probability is greater than 

 0.05 (5%), one can conclude that the ob- 

 servations provide no evidence against the 

 hypothesis. This is an acceptable hypo- 

 thesis. If the probability falls well below 

 0.05 to the 0.01 level or less, the difference is 

 usually considered to be highly significant. 

 As a further example, consider finding 

 6 of one type and 2 of another among a 

 group of 8 individuals. Suppose the the- 

 oretical ratio is 1:1. The probability of 

 obtaining a result this extreme or more 

 extreme according to the null hypothesis is 

 given by computing the sum of the following 

 terms, obtained by expanding i}/i + }/£) • 



Probability of of first type = { l Af 



1 " = 8 X (J/0 8 



2 28 X {Y 2 f 

 " 2 of second type = 28 X (J^) 8 



1 " = 8 X (K) 8 



o « (V 2 ) s 



Adding together these separate exact 

 probabilities, one finds that the total prob- 

 ability of 2 or less of same type = 74/256 

 = 0.29. Since the total probability is 

 greater than 0.05, the statistic is consistent 

 with the hypothesis, which is consequently 

 acceptable. 



