FISHERY BULLETIN: VOL. 85, NO. 2 



METHODS AND AVAILABLE DATA 



The data (Table 1) used for evaluating procedures 

 for estimating the size of the pup population on St. 

 Paul Island were the counts of breeding males made 

 in mid-July and the estimate of the size of the pup 

 population made in early August. 



We assumed that for any year, the sum of the 

 estimated numbers of live pups from each of the 14 

 rookeries, T, was the known or "true" size of the 

 population. Estimates of the variances of each 

 rookery were available; we assumed that the counts 

 from each rookery were independent and estimated 

 the variance of the total population, o^, as the sum 

 of the estimated variances on the 14 rookeries. An 

 approximate 95% confidence interval for the total 

 population was T ± ^(0.975,14) o, where ^(0.975,14) 

 is the 97.5 percentile of Student's t distribution with 

 14 degrees of freedom. 



Two sampling schemes and three estimation pro- 

 cedures were investigated. In particular, estimates 

 based on the standard procedure or the "blow-up" 

 estimate, were compared with ratio and regression 

 estimates, which take advantage of a strong corre- 

 lation between the numbers of breeding bulls and 

 numbers of pups on the rookeries (Figs. 1, 2); these 

 estimation procedures were compared under both 

 simple random sampling and stratified random sam- 

 pling schemes. For each sampling scheme, all possi- 

 ble subsamples of the 14 rookeries were generated, 

 and the distributions of estimates for each of the 3 

 years of data were constructed. To determine how 

 well the estimates predict the "true" population, we 

 computed the fraction of (nominal) 95% confidence 

 intervals about the estimate which contained the 

 "true" value (the actual confidence level of the 95% 



confidence region). In addition, we computed the 

 variance, bias, and the average half-width of the 

 nominal 95% confidence interval of each estimator 

 under the given sampling design. 



"Blow-up" estimates of the total numbers of pups 

 on the rookeries under simple random sampling, 

 T^ij, were calculated in the following way (Cochran 

 1977): Let (Pj, Pg- • •  -P«) be estimates of pup 

 numbers on n sample rookeries. The total on all 

 rookeries was approximated by multiplying the aver- 

 age number of pups on the sample rookeries by the 

 total number of rookeries: 



'^BU - 



14 



1 P. = 14 P. 



n i=i 



(1) 



The estimate of the variance of this estimate is 



Y2.t{Tbu) 



14 



n 



14 



n 



n{n - 1) 1 = 1 



I (P, - P)2 



14 '^ 

 + — 2. Var(P,). 



n 1=1 



(2) 



When sampling was stratified, the above pro- 

 cedure was applied to each stratum. The total num- 

 ber of pups on all rookeries was estimated as the 

 sum of the estimates on all strata; the variance was 

 approximated by applying Equation (2) to each 

 stratum and then summing over the strata. 



Other methods of estimating the total number of 

 pups on all rookeries, when a total count of breed- 

 ing males was available, were suggested through an 

 examination of the regression equations of numbers 



Table 1 .—Numbers of northern fur seal pups counted and their standard deviations, numbers of breeding bulls, and ratio of pups counted 

 to breeding males for the rookeries of St. Paul Island, AK for 1965, 1970, and 1975. 



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