# Statistics for Environmental Engineers

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y = во + в1 (x — ex) + ey = во + Pix + (ey — filex)

In the usual straight-line model, it is assumed that the error in x is zero (ex = 0) or, if that is not literally true, that the error in x is much smaller than the error in y (i.e., ex << ey). In terms of the experiment, this means that the settings of the x values are controlled and the experiment can be repeated at any desired x value. In most calibration problems, it is accepted that the random variability of the measurement system can be attributed solely to the у values. Thus, the model becomes the familiar:

у = Д) + Pi x + ву

Usually the ву are assumed to be independent and normally distributed with mean zero and constant variance a2.

It is now clear that fitting the best straight-line calibration curve is a problem of linear regression. The slope and intercept of the straight line are estimated and their precision is evaluated using the procedures described in Chapter 34. An excellent summary of single component calibration is given by Danzer and Currie (1998).

### Using the Calibration Curve

In practice, the objective is to estimate concentrations (x values) from measured peak areas (у values). The calibration curve is a means to this end. This is the inverse prediction problem; that is, the inverse of the common use of a fitted model to predict у from a known value of x.

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