Sec. 6.3 



ESTIMATION OF ft AND <fi 



163 



at A = 0). (b) Approximately 95 per cent (as nearly as can be told 

 from the graph) of the ti are less than or equal to +2 in magnitude, 

 (c) The middle 80 per cent of the t's with n = 10 fall within the 

 limits —1.5 to +1.5, approximately. Such information will be seen 

 to be needed in arriving at the interval estimate for /x described above. 

 It should be noted that conclusions (b) and (c) of the preceding 

 paragraph referred only to samples with n = 10. The general effect 



1.00 



.90 



& .80 



B 

 a> 



§-.70 



« .60 



_> 



| .50 

 § -40 



a> 



~ .30 



J5 



<S .20 



.10 







-4.0 



•3.0 -2.0 



1.0 



t 



1.0 



2.0 



3.0 



4.0 



Figure 6.32. Relative cumulative frequency distributions for t when n = 5 

 (solid line) and for n = 25 (broken line). 



of the magnitude of n on the frequency distribution of the t's is 

 illustrated in Figure 6.32 for n = 5 and n — 25. The larger the 

 sample size, the less dispersed are the t's. In fact, after n becomes 

 as large as 25 it is difficult to detect much difference between 

 the r.c.f. curve for t and that for a normally distributed measure- 

 ment. Also, if ??i is smaller than n 2 , the ogive for samples with n x 

 observations will be above that for samples with n 2 observations for 

 negative t's and below it for positive t's. This is just a graphic verifi- 

 cation of the fact that the t's are more dispersed for the smaller- 

 sized samples. 



In line with the earlier discussion of degrees of freedom for the 

 estimate of the standard deviation, the t is said to have the same 

 number of degrees of freedom as the standard deviation in the 

 denominator of this ratio. The t's considered so far have one less 

 degree of freedom than the size of the sample, that is, n — 1. 



