530 



AIMM.NDIX 



different -ample t H -a\e 30 ni. ih- .lixl 20 



females. I- there a significant difference 

 in the frequency of males in the two 

 samples? t\ = 0.40 and f B = 0.60; 

 X A = 50, N B = 50.) We have no ex- 

 pectation as to what p x or p B should be. 

 According to the mill hypothesis these two 

 samples have the same parameter, p x . 

 Our besl estimate of p x is f x , obtained by 

 pooling the results of both samples and 

 obtaining 50 loo = 0.50. We next calcu- 

 late li<>\\ large the difference between the 

 observed f's is, relative to the total standard 

 deviation that one would expect if f x were 

 obtained in each of the two samples, N A 

 and N B - This calculation can be made 

 from the expression: 



IB - fA 



V 



fxO -fx) + fx(l -fx) 



X 



N. 



0.20 



\ 



0.5 X 0.5 0.5 X 0.5 



= 2.0 



50 



50 



(The subtraction in the numerator should 

 be made to give a + result, i.e., one should 

 obtain the absolute value of the remainder.) 

 It has been shown that if N x is greater than 

 30, values of 2.0 or more will occur by 

 chance only 5% of the time. We conclude, 

 therefore, that the two samples under test 

 are on the borderline of being statistically 

 different at the 5% level of significance. 



1). The plus-minus test. Suppose a par- 

 ticular treatment is to be tested for its 

 capacity to change a statistic. Suppose, 

 moreover, that one does not care just how 

 much change is being induced as compared 

 with how much is occurring spontaneously. 

 (The treatment might produce only a very 

 small change; under these circumstances, 

 two tremendously large samples, one con- 

 trol and the other treated, would be neces- 

 sary to obtain a statistically significant 

 difference between their measurements.) 



Wli.it can be done i- to arrange a series of 

 paired observations in which the members 

 ot a pair are as similar as possible in order 

 to make the measurement of difference as 

 sensit i\ e as possible. 



Imagine, for example, that one wishes to 

 determine whether feeding a salt to the de- 

 veloping Drosophila male has any effect 

 upon the sex ratio of his progeny. Each 

 test consists of scoring the sex of the prog- 

 eny of two single pair matings, in which one 

 male has and the other has not been treated. 

 Assume that the experiment is performed 

 in an unbiased manner and that the results 

 are as follows: 



Paired 

 Obser- Un 



Sex Ratio 

 (o*oV9 9) 



± Test 



Un- 



vation treated Treated treated Treated 



1 0.47 0.46 + 



2 0.48 0.47 + 



3 0.49 0.48 + 



4 0.50 0.50 No Test 



5 0.46 0.44 -f- - 



6 0.51 0.50 + 



7 0.48 0.47 + - 



One proceeds to test the null hypothesis 

 that the treatment has no effect upon the 

 ¥ x sex ratio. In accordance with this view, 

 there would be an equal chance for the un- 

 treated and treated members of a pair of 

 observations to have the higher sex ratio 

 (that is, to be scored + ); consequently the 

 Ho is p = ]/2- There are only 6 tests of 

 the Ho, since one test gave the same sex 

 ratio for both untreated and treated. The 

 probability that the relevant 6 untreated 

 shall be all successes or all failures is, ac- 

 cording to the null hypothesis, 2(}/£) 6 , or ^ 2 - 

 or about 3%. (The chance that t he remain- 

 ing 5 tests will be like the first is (%) , or 

 also about 3%.) Accordingly, one rejects 

 the null hypothesis at the 5% level of signi- 

 ficance. The statistical test indicates that 



