The desired level of precision is the maximum acceptable error, expressed as a percentage or decimal fraction of the estimated mean. Note that a lower precision level indicates greater precision. A precision level of 10% (0.1) is frequently set as a goal but is seldom achieved.
After a preliminary value for n based on a preliminary value for t* is calculated, the calculation is repeated with the value of t* corresponding to the preliminary value for n.
Equation 10.4(4) assumes that the values for each variable to be measured (in this case the percentages of each solid waste component in the different samples) are normally distributed (conform to the familiar bell-shaped distribution curve, with the most frequent value equaling the mean). In reality, solid waste composition data are not normally distributed but are moderately to severely skewed right, with numerous values several times higher than the mean. The most frequent value is invariably lower than the mean, and in some cases is close to zero. The greater the number of waste categories, the more skewed the distributions of individual categories are.
Klee (1991; 1993) and Klee and Carruth (1970) have suggested equations to account for the effect of this skew-ness phenomenon on the required number of samples. Use of these equations is problematic. Like Equation 10.4(4), they are designed for use with one waste category at a time. For waste categories for which the mean is large compared to the standard deviation, the equations yield higher numbers of samples than Equation 10.4(4). This result is intuitively satisfying because more data should be needed to quantify a parameter whose values do not follow a predefined, normal pattern of distribution. For waste categories for which the mean is less than twice as large as the standard deviation, however, these equations tend to yield numbers of samples smaller than Equation 10.4(4). This result is counterintuitive since no reason is apparent for why an assumption of nonnormal distribution should decrease the quantity of data required to characterize a highly variable parameter.
An alternative method of accounting for skewness is to select or develop an appropriate equation for each waste category based on analysis of existing data for that category. Hilton, Rigo, and Chandler (1992) provide the results of a statistical analysis of the skewness of individual waste categories.
Equation 10.4(4) gives divergent results for different solid waste components. Based on the component means and coefficients of variation shown in Table 10.4.1 and assuming a precision of 10% at 90% confidence, the number of samples given by Equation 10.4(4) is 45 for paper other than corrugated, kraft, and high-grade; almost 700 for all yard waste; and more than 2400 for just grass clippings. The value of Equation 10.4(4) alone as a guide in designing a sampling program is therefore limited.
An alternative method is to estimate the number of samples required to achieve a weighted-average precision level equal to the required level of precision. The weighted-average precision level is the average of the precision levels for individual waste categories weighted by the means for the individual waste categories. The precision level for individual waste categories can be estimated with the following equation, which is Equation 10.4(4) solved for e:
The precision level for each category is multiplied by the mean for that category, and the results are totaled to yield the weighted-average precision level. The number of samples (n) is adjusted by trial and error until the weighted-average precision level matches the required value.
Calculation of the weighted-average precision level is shown in Table 10.4.3 later in this section. Figure 10.4.1 shows the relationship of the weighted-average precision level to the number of samples and the number of waste categories based on the values in Table 10.4.1. Overall precision improves as the number of samples increases and as the number of waste categories decreases. This statement does not mean that studies involving greater number of categories are inferior; it simply means that determining a few things precisely is easier than determining many things precisely.
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