126 SAMPLING FROM BINOMIAL POPULATIONS Ch. 5 



(The central 95 per cent still includes the observed number, r = 10, 

 and it again is possible that p could be smaller and still keep r = 10 

 in the central 95 per cent. Hence, try p = .69.) 



(.31 + .69) 10 = .0000 + .0002 + .0018 + .0108{+ .0422 + .1128 

 (sum = .0128, is <.0250) 



r: [ 1 2 3 4 5 



+ .2093 + .2662 + .2222 + .1100} + .0245 



is <.0250 



r: 6 7 8 9 10 



(The central 95 per cent now just barely excludes the observed num- 

 ber, r — 10; therefore, the smallest value of p which has been con- 

 sidered here and which still keeps r — 10 within the central 95 per 

 cent is .70. However, it is clear that if three decimal places were 

 used, the lower end of the confidence interval would be nearer to 

 .69 than to .70; hence, .69 is taken as the lower end of the 95 per 

 cent confidence interval.) 



To determine the upper end of the 95 per cent confidence interval, 

 it is necessary to find out by a similar procedure how large p can 

 become and still leave the observation, r = 10, in the central 95 per 

 cent of the binomial population with n = 10. Obviously, p can go all 

 the way to 1.00, or 100 per cent, and still not exclude the case when 

 r = 10; hence, p = 1.00 is the upper limit of the 95 per cent confi- 

 dence interval when r has been found to be 10 when n = 10. There- 

 fore, it is concluded that if with n — 10, r is observed to be 10 also, 

 the 95 per cent confidence interval on the true percentage in the 

 population is .69 ^p ^1.00. At the same time, the person doing 

 the sampling is aware that there are 5, or less, chances in 100 that 

 his sample has been sufficiently "wild," or unusual, that it has pro- 

 duced a confidence interval which fails to include the true propor- 

 tion, p, of acceptable products in the population which was sampled. 



The work done above is illustrative of the principles involved but 

 is too laborious to be repeated each time a confidence interval is 

 needed, especially when n > 10. Therefore, advantage is taken of 

 some work done by C. J. Clopper and E. S. Pearson, published in 

 Volume 26 of Biometrika. Table 5.21 was obtained by reading from 

 their graphs the 95 and 99 per cent confidence intervals on p for 

 n = 50, 100, and 250. If n is smaller than 50, the confidence intervals 

 are so wide that they are of doubtful value in practice. However, 



