Statistics for Environmental Engineers

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Many experiments would be improved by answering questions like these during the planning rather than the analysis phase of the experiments.


The simplest case is for a linear function relating the calculated value у to the values of *1, *2, and *3 obtained from measured data. The model has the form:

у =0O + в1 *1 + в2 *2 + в3 *3

where the в’s are known constants and the errors in x1, x2, and x3 are assumed to be random. The expected value of у is:

E(у ) = в0 +в;E(Xj ) + в2Е(X2 ) + вэ£(X3 )

The variance of у is:

Var( у) = e?Var (Xj) + e22Var( x2) + e32Var( x3)

+2e^2Cov( x1 x2) + 2e^3Cov( x1 x3) + 2e2e3Cov( x2 x3)

The covariance terms (Cov(x;, x)) can be positive, negative, or zero. If x1, x2, and x3 are uncorrelated, the covariance terms are all zero and:

Var (у) = e?Var (x1) + e22Var (x2) + e32Var( x3)

The terms on the right-hand side represent the separate contributions of each x variable to the overall variance of the calculated variable у. The derived Var(у) estimates the true variance of у (c2) and can be used to compute a confidence interval for у.

This same approach can be applied to a linear approximation of nonlinear function. Figure 49.1 shows a curved surface у = /(x1, x2) approximated by a Taylor series expansion of у where higher-order terms have been ignored:

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