Statistics for Environmental Engineers

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x


У1


У2


2


2.8


0.44


4


6.2


0.71


6


10.4


0.81


8


17.7


0.93

33.3 Normal Equations. Derive the two normal equations to obtain the least squares estimates of the parameters in y = 0 0+ 01x. Solve the simultaneous equations to get expressions for band b1, which estimate the parameters 00 and 01.

34

Precision of Parameter Estimates in Linear Models


KEY WORDS confidence interval, critical sum of squares, joint confidence region, least squares, linear regression, mean residual sum of squares, nonlinear regression, parameter correlation, parameter estimation, precision, prediction interval, residual sum of squares, straight line.


Calculating the best values of the parameters is only half the job of fitting and evaluating a model. The precision of these estimates must be known and understood. The precision of estimated parameters in a linear or nonlinear model is indicated by the size of their joint confidence region. Joint indicates that all the parameters in the model are considered simultaneously.

The Concept of a Joint Confidence Region


When we fit a model, such as n = во + Pix or n = 0J1 — exp(-02x)], the regression procedure delivers a set of parameter values. If a different sample of data were collected using the same settings of x, different у values would result and different parameter values would be estimated. If this were repeated with many data sets, many pairs of parameter estimates would be produced. If these pairs of parameter estimates were plotted as x and у on Cartesian coordinates, they would cluster about some central point that would be very near the true parameter values. Most of the pairs would be near this central value, but some could fall a considerable distance away. This happens because of random variation in the у measurements.

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