Statistics for Environmental Engineers

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0.566 + 139.759(0.2)± 2.16( 1.093) 1 + 1 + (a2Q ЮЦ1^ = 28.518 ± 2.475


It can be stated with 95% confidence that the interval 26.043 to 30.993 will contain the future single observation recorded at xf = 0.2.

Comments


Exact joint confidence regions can be developed for linear models but they are not produced automatically by most statistical software. The usual output is interval estimates as shown in Figure 34.3. These do help interpret the precision of the estimated parameters as long as we remember the ellipse is probably tilted.


Chapters 35 to 40 have more to say about regression and linear models.

b1


FIGURE 34.3 Contour map of the mean sum of squares surface. The rectangle is bounded by the marginal confidence limits of the parameters considered individually. The shaded area is the 95% joint confidence region for the two parameters and is enclosed by the contour Sc = 15.523[1 + (2/13)(3.81)] = 24.62.

References


Bailey, C. J., E. A. Cox, and J. A. Springer (1978). “High Pressure Liquid Chromatographic Determination of the Immediate/Side Reaction Products in FD&C Red No. 2 and FD&C Yellow No. 5: Statistical Analysis of Instrument Response,” J. Assoc. Off. Anal. Chem., 61, 1404-1414.


Draper, N. R. and H. Smith (1998). Applied Regression Analysis, 3rd ed., New York, John Wiley.

Exercises


34.1 Nonpoint Pollution. The percentage of water collected by a water and sediment sampler was measured over a range of flows. The data are below. (a) Estimate the parameters in a linear model to fit the data. (b) Calculate the variance and 95% confidence interval of each parameter.

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