164 SAMPLING NORMAL POPULATIONS Ch. 6 



Table IV provides an r.c.f. distribution of the sampling ratio, t, 

 for most of the commonly used sizes of samples. This table is in the 

 form of those r.c.f. distributions discussed in Chapter 2. This form 

 is different from that found in most statistical tables, but the form 

 of Table IV fits the purposes of this book better than the traditional 

 table. However, the values in the more usual table can be derived 

 from Table IV quite easily. For example, by Table IV the prob- 

 ability that a random sampling t will have a size below — 2 from a 

 sample of 15(14D/F) is seen to be .033. Because the f-distribution 

 is symmetrical about t = the probability that a t computed with 

 14 D/F will exceed 2 numerically is twice .033, or .066. This is the 

 probability given in the usual table for t = 2 and 14 degrees of free- 

 dom. To obtain such a number as .066 for P in those tables we must 

 interpolate because they give the sampling t's which correspond to 

 specified values of P. 



Table IV will be employed in subsequent discussions instead of the 

 r.c.f. curve because it is both more accurate and more convenient to 

 do so. However, the reader should remember that the two methods 

 are basically the same. The use of tables for the ^-distribution is 

 especially advantageous because there would have to be a different 

 r.c.f. graph for each number of degrees of freedom. 



Suppose, now, that the true population mean, /*,, is not known. In 

 spite of our ignorance of the size of ^ it remains true that sampling 

 values of t will conform to the ^-distribution. For example, for 

 n = 10(9 degrees of freedom), it will be true that 92 per cent of the 

 t's will lie between —2 and +2 (see Table IV). Or, put in terms of 

 a mathematical inequality, it remains true that the following state- 

 ment is correct for 92 per cent of a very large number of samples 

 with n = 10: 



(6.35) -2 < < +2. 



Si 



Approximate empirical verification of the truth of this inequality is 

 found in Table 6.31 above. 



In view of the information just given, the following can be said: If 

 we are about to take a random sample of 10 numerical measurements 

 from a normal population, the probability is .92 that the / for this 

 sample will satisfy inequality (6.35) because 92 per cent of all samples 

 with 9 degrees of freedom do lie within the limits —2 to +2. When 

 the \i is not known this statement still is true but we can compute the 

 t only in terms of the n. To illustrate, suppose that a random sample 



