Sec. 6.7 THE CENTRAL LIMIT THEOREM 185 



Use the G-test to decide if one method tends to produce higher metabolism 

 records than the other, and explain your decision in terms of sampling phe- 

 nomena. Ans. G = 0.100; P > .10; accept // (mi = aO. 



9. Some varieties of wheat produce flour which typically takes longer to mix 

 into proper doughs than others. Decide by the G-test if Kharkof actually has 

 (as appears from the samples) a longer mixing time than Blackhull: 



Kharkof: 3.00, 1.88, 1.62, 1.50, 1.75, 1.38, 1.12, 1.88, 2.50, 1.62, 2.88, 2.50, 3.88, 

 and 2.75. Mean = 2.16. 



Blackhull: 1.25, 2.38, 1.62, 1.50, 1.25, 1.38, 2.25, 2.12, 1.84, 2.38, 2.25, 1.50, 2.00, 

 and 1.62. Mean = 1.84. 



10. Compute the 90 per cent confidence intervals for the two varieties of 

 problem 9 and compare them. Draw appropriate conclusions. 



Ans. CI 90 : - 0.04 ^ | /x x -/i 2 | ^ + 0.69. 



6.7 THE CENTRAL LIMIT THEOREM AND NON- 

 NORMAL POPULATIONS 



The statistical methods which have been discussed in this chap- 

 ter are based on the assumption that the populations involved are 

 normal. In practice this requirement rarely is met rigorously; hence 

 we may wonder if the subject matter of this chapter is chiefly of 

 academic interest because it does not fit actual conditions. This is 

 not the situation because of the truth of the central limit theorem. 



This theorem states essentially that if any population of numerical 

 measurements has a finite mean and variance, n and a 2 , respectively, 

 the frequency distribution of the sampling mean, x, will be essentially 

 a normal distribution with mean = n and variance = a 2 /n if the n is 

 very large. As a matter of fact, the necessary size of n depends on the 

 degree of non-normality of the original population. Tables 6. 71 A, B, 

 C, and D summarize a decidedly non-normal population of counts of 

 flies on dairy cattle, and show some observed distributions of x's for 

 samples with n — 9, 16, and 25. Figure 6.71 displays these same dis- 

 tributions visually. It is rather obvious that none of these sample 

 sizes is very large, and therefore the distributions of x are still notice- 

 ably non-normal. However, the meaning of the central limit theorem 

 is illustrated. 



It can be seen from Tables 6.71 and from Figures 6.71: 



(a) That the parent population is extremely non -normal. 



(b) That even with only nine observations per sample, the distri- 

 bution of x has gone a long step towards fulfilling the ideal expressed 

 by the Central Limit Theorem. 



