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<br />32 <br /> <br /> ^ ^ .:::. ^ mf(Yi - y)2 <br />i YI V(y;) m. Yi = yJm; <br />I <br />1 217.4 1.76 290 0.750 6674.4 <br />2 96.5 4.35 319 0.303 2780.0 <br />3 125.6 0.68 275 0.457 9.63 <br />4 122.6 4.20 271 0.452 19.49 <br />5 143.1 5.19 342 0.418 295.8 <br />6 143.9 2.68 231 0.623 1277.3 <br />7 40.2 2.22 146 0.275 796.4 <br /> 0.46829 <br /> 889.3 <br /> 21.08 <br /> <br />n <br /> <br />Y = L Y In <br /> <br />n <br /> <br />L Yi <br /> <br />n <br /> <br />L V(y;) <br /> <br />n <br /> <br />Lmi <br /> <br />n <br /> <br />L mf(Y - y)2 <br /> <br />1874 <br /> <br />11 853 <br /> <br />(area) of the areas (Appendix 2). We therefore <br />need to know the size (area) of all the N areas in <br />the stratum (not only the total area M as in the <br />previous method), so in practice probability <br />sampling is restricted to streams in which N is not <br />too large. <br />The simplest version is proportional probability <br />sampling with replacement. This means that each <br />section is drawn independently, and that the same <br />section may be included more than once in a <br />sample. This drawback is counterbalanced by the <br />fact that the corresponding sampling without <br />replacement leads to estimators which have to be <br />calculated with computer aid. With replacement, <br />however, the calculations are very simple. With <br />the notation above (M = total stratum area, and <br />Pi = mj /M), an estimator of total population size <br />IS <br /> <br />n <br /> <br />Y = (l/n) L yJpi = mean of yJpi (25) <br /> <br />and the sampling variance V (Y) of Y <br /> <br />^ ^ ^ <br />V(Y) == (l/n)V(y;/p;) <br /> <br />(26) <br /> <br />where V(yJpJ = the (spatial) variance of yJpi' <br />usually calculated as <br /> <br />n <br />'" ^ ^ 2 <br />L. (Y;/Pi - Y) <br />n - 1 <br />Example 7. The following y i and m; values were <br />obtained from 7 sections in a stream, selected by <br />proportional probability sampling with replace- <br />ment. If M = 10000, the result is <br /> <br /> ^ Pi = mJM yJpi <br />1 Yi m. <br />I <br />1 39.0 52 0.0052 7500 <br />2 24.2 80 0.0080 3025 <br />3 58.0 127 0.0127 4567 <br />4 105.3 233 0.0233 4519 <br />5 84.4 202 0.0202 4178 <br />6 109.0 175 0.0175 6229 .;J <br />l <br />7 27.5 100 0.0100 2750 <br />Mean of (yJpJ 4681 l <br />Variance of (y;/p;) 2850797 <br /> ^ <br />Thus, Y = 4681 and ~ <br />SECY) = )2850797/7 = 638 <br />