Sec. 6.3 ESTIMATION OF n AND ff 2 161 



the deviations of the sample X's from their mean, x, the other devia- 

 tion can be computed without any risk of error. If the true mean, n, 

 were known, the n (X{ — n)'s would all be quantities whose specific 

 sizes depended on chance, and a 2 could be estimated with n degrees 

 of freedom, which is one more than s 2 has. Also, the estimate made 

 with ju known would be more reliable than s 2 , a fact which is asso- 

 ciated with its greater number of degrees of freedom. 



As soon as a satisfactory method is available for the estimation of 

 a 2 , it follows that the standard deviation of x — which is o/s/n and 

 is symbolized as <x £ — also can be estimated from the following quantity : 



(6.33) s £ = s/Vn = VS(X - x) 2 /n(n - 1) , 



which is calculated from the observations taken in the sample. It 

 still is true — as for all sampling estimates — that s £ is variable from 

 sample-to-sample. 



Although x is the best specific estimate of the population mean, /x, 

 it is preferable to calculate from the sample an interval in which we 

 can expect the true mean to lie, with a measurable degree of confi- 

 dence in this expectation. The so-called point estimate, x, is almost 

 never exactly right, but an interval can be denned in such a way that 

 we can attach a measure of confidence to the statement that n lies in 

 this interval. This problem can be solved by means of a ratio which is 

 analogous to the (X — \x)/a which was studied in Chapter 4. That 

 ratio involves only one variable, X, and follows the normal distribu- 

 tion. So also does the ratio (x — n)/<r £ . If the standard deviation, 

 a, is not known — which is the usual sampling situation — the corre- 

 sponding ratio 



(6.34) t= (x - n)/ Si 



involves a variable denominator and is not normally distributed. Its 

 degree of departure from normality depends on the size of the sample, 

 n, because the denominator is much less variable for the larger samples. 



Mathematicians have derived a formula for the frequency distribu- 

 tion of the ratio, t, for a sample of any size. Although that derivation 

 is not appropriate to this book, sampling experience will provide an 

 approximation to this distribution, and then mathematical tables will 

 be provided which give the same information more accurately and 

 more easily. 



Table 6.31 presents the frequency and the r.c.f. distributions of 580 

 sampling t's obtained from random samples drawn from the near- 

 normal population of Table 6.21. All samples contained n = 10 



