172 SAMPLING NORMAL POPULATIONS Ch. 6 



show up through an excess of large f's beyond the proportions pre- 

 dictable from Table IV. To be more specific, suppose that attention 

 is centered upon a particular t such that p per cent of the /-distribution 

 lies to the right of this point. Such a point is indicated in Figure 6.41 

 as t p . It is noted from this figure that a much larger fraction of the 

 true t's (figure B) lie to the right of t p than is true for the population 

 resulting from the calculations with the false value for n. This dis- 

 crepancy between the hypothetical and the actual situation will show 

 up in the sampling. Obviously, the greater the discrepancy the more 

 easily it is detected by sampling. 



In practice it is not feasible, efficient, or economical to continue to 

 draw samples from a population until the evidence for or against a 

 certain hypothesis is so overwhelming that there is virtually no doubt 

 of its truth or of its falsity. Instead, it is common to take what is 

 considered an adequate number of observations on the population, 

 choose the risk we shall take of rejecting a true hypothesis, and then 

 reject the hypothesis being tested if t goes beyond that predetermined 

 limit. To illustrate, suppose that a sample of 15 observations is to 

 be taken under conditions which specify the population being 

 sampled, and that it is decided that it is appropriate to take 1 

 chance in 20 of rejecting a true hypothesis. For 14 degrees of free- 

 dom, a t which is at, or above, 2.15 numerically (see Table IV) will 

 occur about 1 time in 20 when the choice of /*, is entirely correct. If 

 we decide to regard all t's which are outside the interval —2.15 ^ t ^ 

 + 2.15 as being the result of a false hypothesis regarding /x, we run 

 a risk of 1 in 20 of rejecting a true hypothesis as a result of sampling 

 variations. 



Problem 6.41. Suppose that some educators test two proposed teaching pro- 

 cedures in the following way : 



(1) All available records and the opinions of teachers are applied to the 

 selection of 20 students who, as a group, do a good job of representing students 

 who will be studying the materials upon which the test is to be based. 



(2) Two equally difficult sections of subject matter are carefully chosen. 



(3) The group of 20 students is taught one section by method A, the other 

 by method B. 



(4) Two equally difficult examinations, one on each section of the subject 

 matter, are formulated by competent teachers and given to the 20 students. 



(5) The average difference, student-by -student, between the two test scores 

 is to be used as the measure of the difference in efficiency between the two 

 methods of instruction. 



