Sec. 6.5 TEST OF HYPOTHESIS #o(m = Ws) 179 



populations of average daily gains. It is concluded that the steers 

 on a diet containing 50 per cent peanut meal will, on the average, 

 produce lower gains than those on only 20 per cent peanut meal. 



Ordinarily the experimenter would wish to carry the statistical 

 analysis farther than this by means of confidence intervals. If the 

 steers on 20 per cent peanut meal do not gain enough more to pay 

 for the added expense of using more of the standard ration which 

 costs more than the peanut meal, it still may not pay to use the diet 

 B. If 95 per cent confidence limits are chosen here, they are deter- 

 mined by the usual methods from 



-2.10 < (0.20 - M )/0.036 < +2.10; or 



0.12 < m < 0.28. 



Therefore, it can be concluded with considerable confidence (asso- 

 ciated with odds of 19 to 1) that the average advantage due to feed- 

 ing 20 per cent peanut meal instead of 50 per cent is at least 0.12 

 pound of gain per day but not over 0.28 pound. Given the current 

 price of steers of the sort under study, we can decide which ration 

 is economically preferable. Obviously, other factors would be con- 

 sidered in practice, but they are separate considerations. 



Although it seems preferable in studies such as those illustrated in 

 this section to have equally many observations in each group, this 

 is not always an attainable goal. If the sample sizes are unequal, 

 say rii and n 2 instead of n each, the above methods are applicable 

 but the formulas are changed to fit these new circumstances. For- 

 mula 6.51 is replaced by 



(6.53) s ;^) + ^) ;and 



\ rii -f- n 2 — 2 

 formula 6.52 is replaced by 



(6.54) s~ d = sVl/m + 1/tia = x — — "-— (1/m + l/n 2 ). 



\ n\ + 7&2 — 2 



Formulas 6.51 and 6.53 are fundamentally the same in all important 

 respects; each is an estimate based on the deviations (Xu — x{) and 

 (X 2 i — x 2 ) in both samples. Likewise, formulas 6.52 and 6.54 are 

 fundamentally alike; each comes from the theorem of mathematical 

 statistics that the variance of the difference between the means of pairs 

 of random samples is the sum of the variances of the two means con- 



