Sec. 5.2 POINT AND INTERVAL ESTIMATION OF p 123 



ance, pq/n, of the estimate p will be quite small for a sample of almost 

 any useful size because the p and the q are each less than unity. This 

 indicates that p will not vary greatly from sample to sample, especially 

 if the sample size is fairly large. As a matter of fact, the size of the 

 variance of p can be made as small as desired by taking the n suffi- 

 ciently large. Hence, this estimate, p, is considered to be a very 

 efficient estimate of p. 



In view of the fact that p is almost always in error to some degree 

 in spite of the fact that it is the best point estimate possible, there are 

 many circumstances in which an interval estimate of p is desirable. 

 The interval estimate also is more difficult to compute and to inter- 

 pret; hence it will be considered in some detail. 



The situation is this: n members of a certain binomial population 

 have been drawn at random so that each member of the population 

 had an equal opportunity to be in the sample, and r of them have 

 been found to have the specified attribute A. Given the proportion 

 r/n observed in the sample, what useful limits can we place on the 

 true proportion, p, of A members in the whole population, and what 

 confidence can we have in those limits? It is customary to call such 

 interval estimates confidence limits, or to say that these limits con- 

 stitute a confidence interval. The degree of confidence which we can 

 place in such limits on p is measured by the probability that the 

 sample has given an interval which actually does include p. As might 

 be expected, this probability is the relative frequency with which the 

 sampling process used will produce an interval which does include p. 

 It will be convenient to use the symbol CI 95 , for example, to designate 

 the confidence interval which has — at the start of the sampling proc- 

 ess — 95 chances out of 100 of including the parameter which is being 

 estimated. 



Suppose that a relatively small manufacturing concern is produc- 

 ing roller bearings which are to be shipped to a larger company 

 manufacturing farm machinery. There will be certain specific stand- 

 ards, such as maximum or minimum limits on diameter, which the 

 bearings must meet before they are considered to be acceptable 

 products. Hence, any large batch of bearings could be grouped into 

 two subgroups marked as "acceptable" and "unacceptable," respec- 

 tively, if every bearing were to have its diameter measured with 

 perfect accuracy. It will be assumed here for simplicity of discus- 

 sion that the company which is to receive the bearings requires that 

 each shipment must be 90 per cent "acceptable" or it can be re- 

 jected. The concern which is producing the bearings will have to 



