Let us consider the frequency histograms for the count of fishermen 

 per hour in the complete census and in Lj^, and make the reasonable assump- 

 tion that they are the same; that is, 



F- f ■ . f • . 



Furthermore, let us assume that 



.*■ 



ri ■ I' °^ "i ■ I ^ (^°' 



i 



C - „» D ^ H ^* DH ^ hi ^ DH ^ ^ii .^ , 



Then K " ^ 2h. «-S-F. "^rS rr— F - — 2. — - (i) 



h d 1 d F 1 dF Fj_ d F^^ ^ ^ 



We can now express Nq and N^ approximately by 



1 



f 



and Nf-^Nu- — 2-ii (3) 



^ H ^ d Fi . 



In order to calculate unbiased variance estimates for the total 

 number of hours of effort, N^j and the total catch, N^, it is necessary 

 to know a great deal more about the sampling distribution of the count 

 of fishermen than is now known. Exact variance estimates could be con- 

 structed, but they would be of doubtful value, due to the artificial 

 nature of the assumptions that would have to be made. Instead, variance 

 estimates are derived which consider only the variation within the 

 complete census information. These variances may be regarded as being 

 minimum, 



f . . 

 Let us now define Xj^ « _ -11, so that N^^ - Z x^. We can rewrite 



d F^ i 



Xi as xi - ^ Ij. 5- fii, where 1!^, - H/feo 

 i 



37 



