116 SAMPLING FROM BINOMIAL POPULATIONS Ch. 5 



so upon the basis of the sample's evidence. If you decide to reject 

 the hypothesis that p — 1/2 whenever the observed number of heads 

 is one of the 4 cases just listed, you will unjustly reject H 1.7 per 

 cent of the time because that is how frequently such cases occur by 

 chance when p does equal 1/2. Nevertheless, some rules for action 

 must be adopted or else nothing can be decided from samples. Hence, 

 it will be supposed that the following rules will be followed after 

 15 sample tosses of the coin in question: 



(a) If r = 0, 1, 2, or 3 heads, you will reject H (p = 1/2) and 

 assert that the coin is biased against heads. 



(b) If r — 4, you will accept H () and play the game on the assump- 

 tion that the coin is not biased against heads. 



These two rules can lead you to correct conclusions and actions, 

 and they also can cause you to make one of two kinds of errors: 



(1) The hypothesis H (p = 1/2), which is being tested by sam- 

 pling, may be rejected when it is true. This will be called an error 

 of the first kind. In the above example, the probability that such 

 an error would occur was noted to be .017 under the rules a and b. 



(2) The H may be accepted when it is false. This will be called 

 an error of the second kind. It should be clear that the likelihood 

 of committing an error of this kind depends on what possibilities — 

 or alternative hypotheses — there are. 



It is customary to set up the hypothesis H in such a way that 

 it is considered more serious to make an error of the first kind than 

 it is to accept a false hypothesis. When this is done the probability 

 of committing an error of the first kind (to be designated by a) is 

 kept low — usually a< .10 — and the rules adopted for acting upon the 

 basis of sampling evidence are chosen so that for a given a the prob- 

 ability that an error of the second kind will be made (to be desig- 

 nated by (3) is as small as possible under the circumstances. 



Referring back to the coin-tossing problem, we see that a = .017. 

 Also, the person who was trying to decide from 15 throws if the coin 

 was seriously biased would not care if p had some size between 1/2 

 and 1/3, but did wish to detect a p as low as 1/3. Hence the alterna- 

 tive hypothesis whose truth could lead to errors of the second kind 

 includes all p's at or below 1/3. For the sake of simplicity it will be 

 assumed that the only alternative hypothesis to H (p = 1/2) is 

 H X (P = 1/3). 



