# 9. APPLICABILITY OF THE METHOD

As already mentioned the parameters a, g and S used to construct the limit curve are only a function of N and of the assumption that the population can or cannot be concave. Thus when the size of a target finite population is known, a random sample of size n will correspond to a predicted pessimistic accuracy computed by means of the function as defined in (7.5). Alternatively, if the population is of infinite size, predicted pessimistic accuracy values will be obtained through the use of the model described by (8.3).

The question arising here is whether the size of a finite population can be known to a reasonable degree of certainty. In most sample-based fishery surveys the population under study is finite and of known size as is, for instance, the case of total number of fishing craft operating from homeports. When the population size varies, then a maximum must be assumed.

The described model also provides information regarding critical sample size and breakpoints in the accuracy growth. As stated in Section 6.2 the critical sample size is when x=0.5 or sample size . It is clear that by simply computing a researcher can immediately determine at which sampling level the accuracy will start a steady increasing process towards 1. However, fixing a sample size by only considering the critical level does not always constitute an optimal approach for the following two reasons:

(a) For small populations the critical sample size and the predicted pessimistic accuracy at critical sample size will not necessarily indicate an expected accuracy of much higher level. This is particularly true in concave populations, such as the total set of recordings of fishing unit activities.

(b) Conversely, and particularly in large populations, an arbitrary selection of a very large sample size well beyond the critical point, may not prove a very cost-effective approach and the user may miss the opportunity of achieving about the same accuracy by using considerably smaller samples.

For these two reasons it is suggested that a table be constructed illustrating the predicted pessimistic accuracy for several sample sizes, so as to obtain more flexibility in the evaluation process of alternative sampling schemes. Table 9.1 and Figure 9.1 give an example of such a tabular approach using a simple electronic worksheet. The tabular data and the diagram refer to a concave population of size N=10 000. The worksheet was programmed to also include the intermediate computational steps and the resulting primary and secondary parameters used for the construction of the pessimistic accuracy curve A_(x).

The presented method also suggests that sampling criteria and practices should be reviewed and adjusted when the original target population is stratified into more homogeneous sub-populations. For instance, if a sample size is known to be effective when applied to a population before stratification, its effectiveness would be reduced if divided proportionally to the size of each of the stratified populations.

Table 9.1

PESSIMISTIC ACCURACY MODEL FOR FINITE POPULATIONS

INPUTTING PARAMETERS

 Please indicate if the population can be concave (=0) or that concave populations should be excluded (=1)
 CONCAVE/NON CONCAVE 0 POPULATION SIZE 10000

Computed model parameters

 Primary parameter W (concave) 0.594502 Primary parameter W (non concave) 0.749925
 Resulting W 0.594502
 Intercept a 0.189041 Intercept g 0.189122 Area S 0.0879589
 Curvature k 0.457405 Coefficient a2 -0.823063 Intercept a1 1.01218
 x=logn/logN Sample size Proportion % ACCURACY(Lower limit) 0 1 0.01 0.189 0.01 1 0.01 0.223 0.02 1 0.01 0.256 0.03 1 0.01 0.287 0.04 1 0.01 0.317 0.05 1 0.01 0.345 0.06 1 0.01 0.373 0.07 1 0.01 0.399 0.08 2 0.02 0.425 0.09 2 0.02 0.449 0.1 2 0.02 0.472 0.11 2 0.02 0.494 0.12 3 0.03 0.516 0.13 3 0.03 0.536 0.14 3 0.03 0.556 0.15 3 0.03 0.575 0.16 4 0.04 0.593 0.17 4 0.04 0.610 0.18 5 0.05 0.627 0.19 5 0.05 0.643 0.2 6 0.06 0.658 0.21 6 0.06 0.672 0.22 7 0.07 0.686 0.23 8 0.08 0.700 0.24 9 0.09 0.713 0.25 10 0.1 0.725 0.26 10 0.1 0.737 0.27 12 0.12 0.748 0.28 13 0.13 0.759 0.29 14 0.14 0.770 0.3 15 0.15 0.780 0.31 17 0.17 0.789 0.32 19 0.19 0.798 0.33 20 0.2 0.807 0.34 22 0.22 0.816 0.35 25 0.25 0.824 0.36 27 0.27 0.832 0.37 30 0.3 0.839 0.38 33 0.33 0.846 0.39 36 0.36 0.853 0.4 39 0.39 0.860 0.41 43 0.43 0.866 0.42 47 0.47 0.872 0.43 52 0.52 0.878 0.44 57 0.57 0.883 0.45 63 0.63 0.889 0.46 69 0.69 0.894 0.47 75 0.75 0.899 0.48 83 0.83 0.903 0.49 91 0.91 0.908 Critical Sample Size 0.5 100 1 0.912 0.51 109 1.09 0.916 0.52 120 1.2 0.920 0.53 131 1.31 0.924 0.54 144 1.44 0.928 0.55 158 1.58 0.931 0.56 173 1.73 0.934 0.57 190 1.9 0.938 0.58 208 2.08 0.941 0.59 229 2.29 0.944 0.6 251 2.51 0.946 0.61 275 2.75 0.949 0.62 301 3.01 0.952 0.63 331 3.31 0.954 0.64 363 3.63 0.957 0.65 398 3.98 0.959 0.66 436 4.36 0.961 0.67 478 4.78 0.963 0.68 524 5.24 0.965 0.69 575 5.75 0.967 0.7 630 6.3 0.969 0.71 691 6.91 0.971 0.72 758 7.58 0.973 0.73 831 8.31 0.974 0.74 912 9.12 0.976 0.75 1000 10 0.977 0.76 1096 10.96 0.979 0.77 1202 12.02 0.980 0.78 1318 13.18 0.981 0.79 1445 14.45 0.983 0.8 1584 15.84 0.984 0.81 1737 17.37 0.985 0.82 1905 19.05 0.986 0.83 2089 20.89 0.987 0.84 2290 22.9 0.988 0.85 2511 25.11 0.989 0.86 2754 27.54 0.990 0.87 3019 30.19 0.991 0.88 3311 33.11 0.992 0.89 3630 36.3 0.993 0.9 3981 39.81 0.994 0.91 4365 43.65 0.994 0.92 4786 47.86 0.995 0.93 5248 52.48 0.996 0.94 5754 57.54 0.996 0.95 6309 63.09 0.997 0.96 6918 69.18 0.998 0.97 7585 75.85 0.998 0.98 8317 83.17 0.999 0.99 9120 91.2 0.999 1 10000 100 1.000

Fig. 9.1 Pessimistic accuracy level