Sec. 5.3 PREDETERMINED HYPOTHESES REGARDING p 131 



volume of data makes it possible to determine, virtually without 

 error, the truth or falsity of many of the hypotheses which play im- 

 portant roles in everyday life and in scientific investigations. In 

 these circumstances samples can be taken and made the basis for 

 satisfactory conclusions. 



The statistical methods needed for a test of a predetermined hy- 

 pothesis regarding some binomial population are intended to decide 

 whether or not it is reasonable (as defined by an accompanying prob- 

 ability statement) to suppose that a given sample actually has been 

 drawn from the binomial population which is specified by the hy- 

 pothesis being tested. In order that such a decision can be made, a 

 basis must be established for comparing a particular sampling result 

 with results to be expected from sampling if the hypothesis being 

 tested is strictly correct. How should this be accomplished? Actu- 

 ally, the problem is a very complex one whose full solution cannot 

 be attempted at the reader's present stage of statistical development ; 

 but some useful and informative rationalizations can be presented. 



Strange as it may seem, a large part of the complexity of this 

 problem comes from the fact that there are so many possible solu- 

 tions that the more difficult job is to choose the best one. This was 

 indicated in the introductory part of this chapter. In that intro- 

 duction a rather simple example was considered and a hypothesis 

 was judged for reasonableness by means of the binomial expansion 

 (q + p) n . It was possible with the aid of that expansion to say 

 that if p = 1/2 only 1.7 per cent of a large number of random sam- 

 ples would have r as small as 0, 1, 2, or 3. The rarity of such occur- 

 rences was made the basis for rejecting H (p = 1/2). Although the 

 risk of falsely rejecting H when p actually is 1/2 is only .017, the 

 likelihood of falsely accepting H when p actually is 1/3 was seen 

 to be high; nearly four chances out of five. It was stated in con- 

 nection with that example that the choice of the best procedure for 

 making decisions from samples depends on this latter probability 

 of an error of the second kind because the probability of an error of 

 the first kind usually is fixed in advance. 



In the example just reviewed, a sampling frequency distribution 

 was employed, and events which fell in the lower frequency intervals 

 — that is, the extreme sizes of r — constituted what is called the 

 region of rejection. On the scale of measurement of r, the points 0, 

 1, 2, and 3 comprised the region of rejection. The general problem 

 of choosing best tests of hypotheses regarding population parameters 

 consists of finding functions of the sample observations and of the 



