124 SAMPLING FROM BINOMIAL POPULATIONS Ch. 5 



inspect its products by means of samples because it is inconceivable 

 that every bearing should be carefully measured. 



Assume that a sample of 10 bearings has been inspected and that 

 all 10 were found to be acceptable. Is this sufficient evidence that 

 the shipment probably is up to the standard? In this connection, 

 consider a binomial population with p only .80; that is, it is well 

 below the standards set above. The probability that every member 

 of a sample of 10 will be acceptable is (.80) 10 , which is .11; hence 

 there is about 1 chance in 9 that this definitely substandard batch 

 of bearings will show none unacceptable on a sample of but 10. Ob- 

 viously, if p were less than .80, p 10 would be less than .11; and, con- 

 versely, if p were larger than .80, p 10 would be greater than .11. 

 Therefore, it should be clear that the result, 10 acceptable bearings 

 out of 10 inspected in a sample, could be obtained from any one of 

 a whole range of possible binomial populations corresponding to 

 values of p ranging from to 1. As a matter of fact, the sample 

 discussed above could be drawn at random from any binomial pop- 

 ulation with as many as 10 acceptable bearings among the individ- 

 uals. Of course, with n = 10, a sample with r also equal 10 is more 

 likely to come from a population with p near 1 than from a popula- 

 tion with p near 0. 



The above discussion re-emphasizes the fact that we cannot attain 

 certainty in conclusions drawn from samples: there always must be 

 some risk that the sample has led to a false conclusion. We choose 

 a risk of error which we can afford to take and express it in terms 

 of the confidence coefficient described earlier. If it be supposed that 

 an event which is as unlikely to occur as 1 time in 20 can be ignored, 

 what confidence interval (CI 95 ) can we set on p as a result of the 

 above sample in which r = 10 acceptable bearings out of 10 observed 

 in the sample? 



We use what will be called a central 95 per cent of all possible r's 

 by determining a range on r which is such that not more than 2% 

 per cent of all samples with the same n and p will fall beyond each 

 end (separately) of the range so determined. For example, in the 

 series below for (1/4 + 3/4) 10 the first five terms — to the left of 

 the brace — add to .0197, which is less than 2% per cent, or .0250. If 

 the sixth term from the left is added, the sum exceeds .0250. There- 

 fore, among all possible samples of 10 observations from a binomial 

 distribution with n = 10 and p = 3/4 the sample number, r, will be 

 below 5 for a bit less than 2% per cent of all such samples. At the 

 other end of the series for (1/4 -f 3/4) 10 no term is less than or equal 

 to .0250; hence, the "central 95 per cent" will be occupied by samples 



