Elementary Biometrical Inferences 



537 



then the 95% confidence interval for m can 

 be determined as follows. Given, as before, 

 thatX = 68.03, s x = 3.24, and N = 9; first 

 find the value of t which has p = 0.05 for 

 N — 1 degrees of freedom. (For N — 1 = 8, 

 this is about 2.3.) Hence, 



The value of t is then found from 



X - 



4 



= 2.3 



N 



One rejects all values where X differs 

 from n by more than 2.3 Sj, and accepts all 

 values of X — m with 95% confidence that 

 are less than 2.3 s x . Substituting, one finds 



X-, 



1.08 



= 2.3 



or, X - n = 2.3 (1.08) = 2.48. Finally, 

 the 95% confidence level for n, in the 

 present case, is X ± 2.48, or 65.55 to 70.51. 



d. Comparison of X x and X 2 . Suppose 

 one selects two sample sets of corn plants, 

 and then measures the height of each plant. 

 The statistics obtained are: 



Sample 1: N x = 9 X, = 72.44 



2(X, - X t ) 2 = 65.70 



Sample 2: N 2 = 10 X 2 = 70.30 



2(X 2 - X 2 ) 2 = 69.50 



To be tested is the null hypothesis that 

 these two samples have the same n and 

 the same a x . The best estimate of the 

 unknown a x is s x , obtained from the two 

 samples by the following formula: 



Sx 



. /Z (Xi - X t ) 2 + S(X 2 - X 2 ) : 

 (Ni - 1) + (N 2 - 1) 



-4 



65.70 + 69.50 



+ 9 



= 2.82 



One derives a value of s- in the present 

 case equal to 1.29, since it is known that 





N, ^N 2 



Xi — X.} 



to be 



72.44 - 70.30 

 1.29 



= 1.66. 



Since each X was obtained from a single 

 sample, the number of degrees of freedom 

 is (N t - 1) + (N 2 - 1), or 17. Because 

 p > 0.1 one accepts the null hypothesis 

 and may conclude that the two means are 

 not statistically different at the 5% level 

 of significance. If one obtains a value of t 

 inconsistent with the hypothesis, the two 

 samples differ either in their ju's, ff x 's, or 

 both. 



IV. THE POWER OF THE TEST 



There are two types of error involved in 

 testing a parameter or statistic. One has 

 already been discussed. This type of error 

 is the rejection of the correct hypothesis 

 5% of the time (when working at the 95% 

 confidence level, or the 5% level of sig- 

 nificance) in order to reject incorrect hy- 

 potheses. The other type of error is the 

 incorrect acceptance of an hypothesis. 

 Suppose f = 0.45 and N = 100. The hy- 

 pothesis that p = 3^ is tested and found 

 acceptable at the 5% level. But the real p 

 might lie anywhere between 0.35 and 0.55 

 (see Figure A-2). If p is not 0.5 but some- 

 where between 0.35 and 0.55, one may have 

 accepted the wrong hypothesis. 



In the present case, the test is only power- 

 ful enough to reject incorrect hypotheses 

 where p < 0.35 or > 0.55. Had N been 1000 

 and f = 0.45, the discriminatory power of 

 the test would be greater, at the 5% level, 

 causing the rejection of any hypothesis 

 where p < 0.42 or > 0.47. Before collect- 

 ing statistics, it is necessary to determine 

 whether there is adequate power to dis- 

 criminate against alternative hypotheses. 



Suppose, for genetic reasons, one wishes 

 to test whether some statistics obtained by 

 experiment exhibit an expected 3 : 1 ratio. 

 One may accept the hypothesis; but if a 



