Sec. 6.3 ESTIMATION OF M AND o°- 169 



tive frequency with which the confidence interval will include the 

 parameter. Basically, however, the methods of these two chapters 

 involve the same kind of statistical inference. 



It may have occurred to the reader to wonder why the confidence 

 interval is taken in the center of the sampling distribution. Al- 

 though it is true that 92 per cent of all sampling t's with 9 degrees 

 of freedom will have sizes between — 2 and +2, it is also true that 

 92 per cent of all sampling t's with 9 degrees of freedom will lie be- 

 tween — 5 and +1.54 (see Table IV). Therefore, the inequality 



x — u 

 -5.0 < < +1.54 



Si 



also will be true for 92 per cent of all samples with 9 degrees of free- 

 dom. Why not use this inequality as the basis for computing the 92 

 per cent confidence interval instead of the one suggested earlier? 

 Suppose the inequality above is used on the example used previously 

 in which x = 8 and s £ — 2. The 92 per cent confidence interval now 

 is from 5 to 18 instead of the shorter interval, 4 to 12, obtained previ- 

 ously. It always will be longer when a non-centrally located interval 

 on t is used. It should be clear that the shorter the confidence interval 

 for a given confidence coefficient, the better the interval estimate. 

 Why be more indefinite than is necessary? 



PROBLEMS 



1. Verify the 80, 90, and 95 per cent confidence intervals given in Table 6.32 

 for samples 1 and 2. 



2. Compute 99 per cent confidence limits on /i for samples 8 and 9 of Table 

 6.32 and interpret them. Arts. 57.2 ^ fi ^ 78.2. 43.2 ^ /x ^ 64.8. 



3. Given that .f = 35 and s = 10 for a sample of ten observations compute and 

 interpret the 95 per cent confidence interval on /*. Do the same for n = 15 and 

 n = 20 and compare them. What is the implication regarding the relation be- 

 tween the size of the sample and the width of the confidence interval, every- 

 thing else being equal? 



4. Given that the t was computed to be —2.08 for sample number 525, Table 

 6.32, determine whether or not the 90 per cent confidence interval includes /i. 

 Do likewise for 95 per cent limits. 



Ans. CI 90 does not include /x = 60; CI 95 does. 



5. Use Figure 6.31 to determine the 86 per cent confidence interval on /j. 

 from sample 6 of Table 6.32. 



6. Suppose that an improved method of cultivating wheat has produced an 

 average of 5 bushels per acre more yield than an older method on a sample of 

 21 plots. Also assume that the standard deviation on this sample is s = 5 

 bushels per acre. What are the 95 per cent confidence limits on the true aver- 



