Statistics for Environmental Engineers

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b ± tvia/2SE(b)


For a = 0.05, v = 5, we find t5 0025 = 2.571 , and the 95% confidence limits are 0.1 ± 2.571(0.0018) =


0.1 ± 0.0046.


Figure 33.4a expands the scale of Figure 33.3a to show more clearly the confidence interval computed from the t statistic. The sum of squares function and the confidence interval computed using the t statistic are both symmetric about the minimum of the curve. The upper and lower bounds of the confidence interval define two intersections with the sum of squares curve. The sum of squares at these two points is identical because of the symmetry that always exists for a linear model. This level of the sum of squares function is the critical sum of squares, Sc. All values of в that give S < Sc fall within the 95% confidence interval.


Here we used the easily calculated confidence interval to define the critical sum of squares. Usually the procedure is reversed, with the critical sum of squares being used to determine the boundary of the confidence region for two or more parameters. Chapters 34 and 35 explain how this is done. The F statistic is used instead of the t statistic.


Linear Model


Nonlinear Model


0.10.


ш я


0.05-


o


E


я


0.00


0.03-|


Г-T


0.02


Sc=0.0175


Sc=0.027


0.01-


00 i


Г—~ 1 1 CM


T-.: : cm d; ;o

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