# Statistics for Environmental Engineers

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—    0.0050

—    -0.10

Ak 1.00 -0.90 AX     20

Ak 0.70-0.90 AA2    2

These sensitivity coefficients can be used to estimate the expected variance of k:

— (0.0050) 2100 + (-0.10 )21 — 0.0025 + 0.010 — 0.0125

and

at — 0.11

An approximate 95% confidence interval would be k — 0.90 ± 2(0.11) — 0.90 ± 0.22, or 0.68 < k < 1.12.

Unfortunately, at these specified experimental settings, the precision of the estimate of k depends almost entirely upon X2; 80% of the variance in k is contributed by X2. This may be surprising because X2 has the smallest variance, but it is such failures of our intuition that merit this kind of analysis. If the precision of k must be improved, the options are (1) try to center the experiment in another region where variation in X2 will be suppressed, or (2) improve the precision with which X2 is measured, or (3) make replicate measures of X2 to average out the random variation.

### Propagation of Uncertainty in Models

The examples in this chapter have been about the propagation of measurement error, but the same methods can be used to investigate the propagation of uncertainty in design parameters. Uncertainty is expressed as the variance of a distribution that defines the uncertainty of the design parameter. If only the range of parameter values is known, the designer should use a uniform distribution. If the designer can express a “most likely” value within the range of the uncertain parameter, a triangular distribution can be used. If the distribution is symmetric about the expected value, the normal distribution might be used. The variance of the distribution that defines the uncertainty in the design parameter is used in the propagation of error equations (Berthouex and Polkowski, 1970).

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