Statistics for Environmental Engineers

Скачать в pdf «Statistics for Environmental Engineers»


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

Theory


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:

Скачать в pdf «Statistics for Environmental Engineers»