Sec. 6.6 THE G-TEST WITH SMALL SAMPLES 183 



instead of Table IX. Again the tables give P(\ G \ > Go), where G 

 is the observed numerical size of G. 



To illustrate the application of formulas 6.61 and 6.62 reference is 

 made to the problems solved in sections 6.4 and 6.5. First consider 

 problem 6.41. The sample mean is x = 5.65 and the range is 15; 

 therefore, for # (m = 0): G = (5.65 - 0)/15 = 0.377, with the sam- 

 ple size = n = 20. By Table IX, the probability that G would be so 

 large if /x actually were zero is much less than .001 ; hence the hypothesis 

 that At = is rejected decisively, as it was from the £-test. 



The next example is from section 6.5 and involves two diets fed to 

 steers. In fact, x x = 1.54 and x 2 = 1.74, R x = 0.26, R 2 = 0.29, and 

 hence the average sample range = 0.275. Then 



G = 0.20/0.275 = 0.727, with each n = 10. 



By Table X a G larger than 0.727 would occur by chance less than 

 0.1 per cent of the time if both samples were from the same normal 

 population. The hypothesis is rejected; that is, the second diet, 

 which produced the higher average gain in the sampling is considered 

 to produce higher average gains than the first diet. 



Given the tables and formulas above, we can derive confidence 

 intervals on /x as before when n is small. This interval would not 

 be expected to be identical with one obtained from the ^-distribution 

 for the same confidence coefficient; but it has been shown that, on 

 the average, the two intervals are very close to the same length as 

 long as n is small. (Specifically, K. S. C. Pillai has shown in the 

 September, 1951, number of the Annals of Mathematical Statistics 

 that the ratio of the average lengths of the CI 95 's by the two meth- 

 ods still is 0.97 when n = 20.) To illustrate, consider again the 

 problem of section 6.4 just used above to illustrate the G-test when 

 there is one set of n observations. In this problem the two confi- 

 dence intervals are obtained as follows: 



5.65 — a 



-2.1 < < +2.1 



~ 0.87 " 



and the 95 per cent confidence interval is 



3.82 <n< 7.48. 



Using the ratio G, we have 



5.65 — n 



-0.126 < < +0.126 



15 



