Statistics for Environmental Engineers

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The assessment given in Example 2.9 can also be made by examining the reference distributions of y . The distribution of y is centered about n = 8.0 mg/L with standard deviation s = 0.266 mg/L. The value of y observed for this particular experiment is 7.51 mg/L. The shaded area to the left of y = 7.51 in Figure 2.12(a) is the same as the area to the left of t = -1.853 in Figure 2.12(b). Thus, P(t < -1.853) = P( y < 7.51) « 0.04.

In the context of Example 2.9, the investigator is considering the particular result that y = 7.51 mg/L in a laboratory assessment based on 27 blind measurements on specimens known to have concentration П = 8.00 mg/L. A relevant reference distribution is needed in order to decide whether the result is easily explained by mere chance variation or whether it is exceptional. This reference distribution represents the set of outcomes that could occur by chance. The t distribution is a relevant reference distribution under certain conditions which have already been identified. An outcome that falls on the tail of the distribution can be considered exceptional. If it is found to be exceptional, it is declared statistically significant. Significant in this context does not refer to scientific importance, but only to its statistical plausibility in light of the data.

Significance Tests

In Example 2.9 we knew that the nitrate population mean was truly 8.0 mg/L, and asked, “How likely are we to get a sample mean as small as y = 7.51 mg/L from the analysis of 27 specimens?” If this result is highly unlikely, we might decide that the sample did not represent the population, probably because the measurement process was biased to yield concentrations below the true value. Or, we might decide that the result, although unlikely, should be accepted as occurring due to chance rather than due to an assignable cause (like bias in the measurements).

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