Sec. 5.1 OBTAINING THE SAMPLE 117 



The p can be determined from the following series: 



(2/3 + 1/3) 15 = .002 + .017 + .060 + .130 + .195 + .214 + .179 

 r: 1 2 3 4 5 6 



+ .115 + .057 + .022 + .007 + .002 + .000 + .000 

 r: 7 8 9 10 11 12 13 



+ .000 + .000. 

 r: 14 15 



Hence if p actually is 1/3 so that the hypothesis of no bias should 

 be rejected, the probability is .002 + .017 + .060 + -130 = .209 that 

 H will be rejected. Or the probability that H will not be rejected 

 when it should be — an error of the second kind — is (3 = 1 — .209 = 

 .791. Obviously, the rules a and b would not be good ones if it is 

 serious to fail to detect the bias indicated by p = 1/3. However, if 

 the most serious mistake is to accuse someone of employing a biased 

 coin when he is innocent, rules a and b may be quite satisfactory. 



In practice we seldom can compute (3 as simply as above. Usually 

 the a is set at an appropriate level and then standard tests are em- 

 ployed without actually knowing the p. However, it can be said 

 here that the tests to be discussed in this, and the next, chapter have 

 been chosen with the idea of making the p as small as possible under 

 the circumstances and for the chosen a. 



As the heading of this chapter indicates, the subsequent discussion 

 will be confined to samples from binomial populations. Later chap- 

 ters will take up the normal and the two-variable situations. 



5.1 OBTAINING THE SAMPLE 



Before a method for obtaining the sample is devised, the popula- 

 tion which is to be sampled must be defined clearly. It is recalled 

 from Chapter 4 that a binomial population is possible only if the 

 units in some definable group have attributes which may be described 

 by just two classes. Moreover, the fractional part of the population 

 falling into each class must stay fixed. For example, all the farmers 

 in Finney County, Kansas, on July 1, 1953, could be classified un- 

 ambiguously into two classes as regards membership in some co- 

 operative association: those who do belong to some cooperative and 

 those who do not belong to any such association. The units would 



