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FISHERY BULLETIN OF THE FISH AND WILDLIFE SERVICE 



to various values of sample tag ratio ?*="• Since 



to every value of p there corresponds a value of A'', 

 these curves also give the confidence limits of 

 our estimate of the size of the population made by 

 the formula 



T 



(14) 



T 



V 



which is the same as formula 1, of course. 



A niunerical example may make this clear. 

 Suppose that in a given experiment we have 

 placed 1,000 tagged fish in the population and 

 plan to draw a sample of 2,000 fish for determinmg 

 the tag ratio. By formula 13 we can compute for 

 values of population tag ratio, p, the limits within 

 which p* will be expected to fall in, say, 95 percent 

 of the cases (e=0.95). In figure 1, we have 

 calculated and plotted these limits for part of the 

 range of p for this example. The ordinates on 

 this graph are values of p, and the abscissae are 

 values oi p*. Going horizontally across the graph 

 for a given value of p we come to the values of p* 

 withm which samples of 2,000 from a population 

 having a true tag ratio of p would be expected to 

 fall in 95 percent of the cases. By the theory 

 developed by Neyman the loci of such points for 

 various values of p form the 95-percent confidence 

 limits for values oi p*. For a given value oi p* 

 we go along the vertical to the intersections with 

 these loci to find the confidence limits for that 

 value of p*. Thus, suppose that we draw our 

 sample of 2,000 and find that it contains 100 

 tagged fish. Our estunate of the tag ratio in the 

 population is 0.05, and from figure 1 we find that 

 for this value oi p* the 95-percent confidence 

 lunits are 0.042 and 0.059. Since we know there 

 are 1,000 tagged fish in the population, our 

 estimate of the population by formula 14 is 20,000 

 with 95-percent confidence limits 16,950 and 

 24,800. On the right-hand edge of the graph we 

 have plotted the values of A^^ corresponding to 

 tag-ratio values of the same ordinates on the 

 left-hand edge, in order to exhibit graphically the 

 relation between the two. 



Such a chart as this may be computed for any 

 particular experiment . The entire range of values 

 of p need not be included; it is sufficient in practice 

 to compute the values to include the region within 

 which p* \a expected to fall. 



For values of n which are small with respect to 

 N, so that -j^ — T approaches 1, formula 13 ap- 

 proaches the form appropriate for the binomial. 

 Clopper and Pearson (1934) have computed and 

 charted the confidence limits of the binomial for a 

 large number of values of n for 95 percent and 

 99 percent confidence levels. Since the limits for 

 the binomial fall in every case outside the limits 

 given by formula 13, these charts may be used to 

 obtain upper and lower limits on the sample value 

 oi p* even where n is not small in relation to A'^. 

 This involves, of course, a considerable loss of 

 efficiency when n is not small in relation to A'', so 

 that the employment of formula 13 would seem to 

 be generally preferable in such cases. 



Chapman (1948) has considered the Poisson 

 approximation to the distribution of expected 

 numbers of tag recoveries where the tag ratio is 

 low, in addition to the normal, normal-binomial, 

 and normal-hypergeometric approximations, as 

 bases for confidence-interval estimates of A^. He 

 has tabulated useful confidence limits for the 

 Poisson distribution, and discusses practical cri- 

 teria for judging which distribution to choose as a 



basis of estimation for various values of n and — 



n 



As is shown by Chapman's example on page 81 

 of his paper, for experiments involving numbers of 

 tagged fish, T, and subsequent samples, n, of the 

 magnitude of the example we have employed, and 

 which is of the approximate magnitude of most 

 practical tagging experiments, the differences in 

 confidence lunits resulting from the several dis- 

 tributions which might be employed are not very 

 great. In practice it would make little difference 

 which we chose. He recommends which distribu- 

 tion to employ for various situations ; for values of 



n> 1,000 and->0.05he recommends the normal 



hypergeometric, which has been employed by me 

 in the example above. 



REPEATED SAMPLING OF A CONSTANT 

 POPULATION 



Where the population of an area remains con- 

 stant over an appreciable period of time, it is 

 possible to arrive at an estimate based on repeated 

 sampling and marking. 



