ELEMENTARY SAMPLING THEORY FOR ENGINEERING 27 



the sample, then, two per cent of the relays were defective. What, 

 then, is the probability that the percentage of the 4,000 relays having 

 this defect is between one and three per cent? Or what is the proba- 

 bility that the percentage of defectives in the universe of 4,000 does not 

 exceed four per cent? Or again, if we wish to be practically certain that 

 among the 4,000 relays not more than two per cent are defective in 

 this respect, how many defectives would be allowable in a sample of 

 200? or a sample of 1,000? Any number of questions of this sort 

 can be asked and may be answered on the basis of the proper assump- 

 tions by sampling theory. 



Example 2: Sampling of Variables 



An office serves 5,000 subscribers lines. Measurements of the 

 insulation resistance are made on 200 of these, selected at random, 

 and the resulting values tabulated. They vary all the way from 

 12,000 ohms to 200,000 ohms. What conclusions may be drawn as 

 to the probability that more than a certain number, say 20 of the 

 subscribers' loops out of the 5,000, have an insulation resistance of 

 less than 18,000 ohms? What is the most probable distribution of 

 the insulation resistances for the office as a whole? What is the 

 probable error of the average of the observations as a measure of the 

 average loop insulation resistance for the office? 



As before, much information regarding the universe may be inferred 

 from a properly chosen sample, always, however, with some degree 

 of uncertainty. This uncertainty, so far as the sampling process is 

 concerned, naturally decreases as the size of the sample increases, 

 and, of course, disappears except for inaccuracies of measurement, 

 when the sample becomes coextensive with the universe. 



The respective treatments of these two types of problems differ 

 considerably in detail. The basic principles are, however, essentially 

 the same, and involve in each case the notions of "a posteriori" 

 probability, as discussed in most of the standard textbooks on the 

 theory of probability. 



In both problems there are certain observations. By means of 

 these we desire to obtain as precise information as possible concerning 

 some one or more characteristics of the universe from which these 

 observations or samples were drawn. The true nature of the universe 

 is to some degree, at least, unknown. Certain hypotheses concerning 

 it may, however, in the light of the sample be more probable than 

 others. What we wish to estimate is the probability that either a 

 particular hypothesis or a group of mutually exclusive hypotheses 

 includes the true one. 



