Statistics for Environmental Engineers

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An interesting problem arises in using the calibration curve in this inverse fashion. Of course it is a simple matter to use the fitted equation (у = b0+ b1x) to compute x for any given value of у. Specifically, the estimate of x for any future observation у is x = (у — b0 )/b1. Because the calibration curve is to be used over and over again, what is required for a future value of у, in addition to the predicted value x, is an interval estimate that has the property that at least 100P% of the intervals will contain the true concentration value | with 100(1 — a)% confidence. For example, for P = 0.9 and a = 0.05, we can assert with 95% confidence that 90% of the computed interval estimates of concentration will contain the true value. This problem was resolved by Leiberman et al. (1967). Hunter (1981) gives details.


The error in x clearly will depend in some way on the error in measuring у and on how closely b0 and b1 estimate the true slope and intercept of the calibration line. The uncertainties associated with estimating P0 and p1 imply that the regression line is not unique. Another set of standards and measurements would yield a different line. We should consider, instead of a regression line, that there is a regression band (Sharaf et al., 1986), which can be represented by a confidence band for the calibration line. The confidence band for the entire calibration line (i.e., valid for any value of x) can be used to translate the confidence interval for the true response n, based on a future observation of у, into a confidence interval for the abcissa value.

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