Statistics for Environmental Engineers

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For the one-parameter linear model through the origin, the variance of b is:


Var( b) = §-2 L x

The summation is over all squares of the settings of the independent variable x. a2 is the experimental error variance. (Warning: This equation does not give the variance for the slope of a two-parameter linear model.)

Ideally, a 2 would be estimated from independent replicate experiments at some settings of the x variable. There are no replicate measurements in our example, so another approach is used. The residual sum of squares can be used to estimate a2 if one is willing to assume that the model is correct. In this case, the residuals are random errors and the average of these residuals squared is an estimate of the error variance o2. Thus, a2 may be estimated by dividing the residual sum of squares (SR) by its degrees of freedom (v = n p), where n is the number of observations and p is the number of estimated parameters.

In this example, SR = 0.0116, p = 1 parameter, n = 6, v = 6 — 1 = 5 degrees of freedom, and the estimate of the experimental error variance is:

s2 = __Sr_ = °mi6 = 0.00232

n p    5

The estimated variance of b is:

Var (b) =2 = °00jf32 = 0.0000033 Ex;    713

and the standard error of b is:

SE (b) = VVar( b) = 70.0000032 = 0.0018 The (1-a)100% confidence limits for the true value в are:

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