# Statistics for Environmental Engineers

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Example 8.2

A sample of observations, y, appears to be from a lognormal distribution. A logarithmic transformation, x = ln( y), produces values that are normally distributed. The log-transformed values have an average value of 1.5 and a standard deviation of 1.0. The 99th quantile on the log scale is located at z099 = 2.326, which corresponds to:

x099 = 1.5 + 2.326( 1.0) = 3.826.

The 99th quantile of the lognormal distribution is found by making the transformation in reverse:

yp = antilog (xp) = exp (xp) = 45.9.

An upper 100(1-a)% confidence limit for the truepth quantile, yp, can be easily obtained if the underlying distribution is normal (or has been transformed to become normal). This upper confidence limit is:

UCLj_a( Ур) = y + K 1-a,pS

where K1-ap is obtained from a table by Owen (1972), which is reprinted in Gilbert (1987).

Example 8.3

From n = 300 normally distributed observations we have calculated y = 10.0 and s = 1.2. The estimated 99th quantile is y0.99 = 10 + 2.326(1.2) = 12.79. For n = 300, 1 — a = 0.95, and p =

0.99, K095099 = 2.522 (from Gilbert, 1987) and the 95% upper confidence limit for the true 99th percentile value is:

UCL0.95( y0.99) = 10 + (1.2)(2.522) = 13.0.

In summary, the best estimate of the 99th quantile is 12.79 and we can state with 95% confidence that its true value is less than 13.0.

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