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ATTACHMENT D SAMPLING METHODOLOGY AND CALCULATIONS <br />where: <br />Ci <br />Wi <br />i <br />c = weight of particular component <br />w = sum of all component weights <br />for i 1 to n <br />where n = number of selected samples <br />for j 1 to m <br />where m = number of components <br />The low and high, or confidence interval, for this estimate is derived from a nonparametric <br />statistical technique called the Bootstrap (Efron, B. 1982. The Jackknife, the Bootstrap, and other <br />Resampling Plans. Society for Industrial and Applied Mathematics). Standard methods of <br />calculating sample statistics are generally not applicable to waste composition results <br />because each substream consists of multiple waste components that must sum to one for <br />each substream. The distribution of these components is a multinomial with unknown <br />properties. As such, sample statistics other than the sample mean proportions cannot be <br />calculated using standard parametric techniques without making unappealing assumptions <br />that would invalidate the results. <br />The Bootstrap method is a simulation technique that allows the calculation of the variance <br />and other statistics of a parameter with unknown distributional properties. In this study, the <br />Bootstrap method was used to calculate the square root of the Bootstrap variance estimates <br />of each sample mean (henceforth referred to as the standard error). The mean and standard <br />error were then used to calculate confidence intervals about sample mean estimates. <br />The upper and lower confidence limits provide the boundaries of an interval within which <br />we are 90 percent confident that the true mean proportion of a waste type will lie. They <br />represent the high and low estimates shown in this study. <br />Upper and lower confidence limits were calculated as follows: <br />Cl. = SMg + (1.645 *SEg) <br />CI1= SMg - (1.645 *SEg) <br />where: Cl. = upper confidence limit <br />Ch = lower confidence limit <br />SMg = sample mean proportion for waste component g <br />1.645 = standard normal deviate (two- tailed) at a 0.05 level <br />D -4 APPX B WASTE COMPOSITION REPORT 121709.DOC <br />