Sec. 6.4 



TEST OF HYPOTHESIS H (n = /*o) 



171 



sampling results, that hypothesis should be rejected. However, if 

 the evidence in the sample is in reasonable accord with that hypothe- 

 sis, it should be accepted. This is the idea behind the methods to 

 be presented in this section, and also of all so-called tests of signifi- 

 cance. 



A generally satisfactory solution to the problem of this section can 

 be obtained from the ^-distribution when normal or near-normal popu- 

 lations are being sampled. As the reader already knows, t = (x — m)Ax 

 and has n — 1 degrees of freedom if the sample contains n observa- 

 tions. When the m is specified by the hypothesis to be tested, the t 

 can be calculated. Thereafter, 'we can determine from Table IV how 



Figure 6.41. Illustration of the effects on the (-distribution of a false hypothesis 



regarding /x. 



uncommon such a t is when the samples are drawn from the supposed 

 population. For example, if n = 14 and t turns out to be 0.90 on the 

 assumption that n = 0, we learn from Table IV that about 38 per cent 

 of all sampling t's with 13 degrees of freedom are numerically larger 

 than 0.90. Therefore, this value of t is not at all unusual and hence 

 we would have no reason to doubt the hypothesis that y. = 0. But, 

 suppose that t had been 3.0. It is seen in Table IV that only about 

 one t in 100 from samples of this size ever gets as large as 3, numeri- 

 cally. Hence, we might reasonably doubt that n really is zero because 

 t rarely attains such a size when the hypothesis being tested is true. 



To illustrate the above discussion graphically, suppose that the true 

 mean, ju, of a normal population of measurements actually is 2 but 

 owing to some error in reasoning jj. is considered to be 0. What effect 

 does this have on the frequency distribution of VI For this situation, 

 t really is {x — 2)/s £ but because of the error regarding /z the values of 

 t are calculated from the formula t = x/s £ . In view of the fact that 

 t\ — (x — 0)/s £ is just 2/s-e units larger than t 2 = (x — 2)/s if we are 

 actually sampling population B of Figure 6.41, but we think that we 

 are sampling from population A. The discrepancy should, and would, 



