Statistics for Environmental Engineers

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Confidence Intervals


Hypothesis testing can be overdone. It is often more informative to state an interval within which the value of a parameter would be expected to lie. A 1 — a confidence interval for the population mean can be constructed using the appropriate value of t as:


У — syta/2 < П < У + syta/2


where ta/2 and sy have v = n — 1 degrees of freedom. This confidence interval is bounded by a lower and an upper limit. The meaning of the 1 — a confidence level is “If a series of random sets of n observations is sampled from a normal distribution with mean n and fixed a, and a 1 — a confidence interval y ± sy ta/2 is constructed from each set, a proportion, 1 — a, of these intervals will include the value and a proportion, a, will not” (Box et al., 1978). (Another interpretation, a Bayesian interpretation, is that there is a 1 — a probability that the true value falls within this confidence interval.)


Example 2.12


The confidence limits for the true mean of the test specimens are constructed for a/2 = 0.05/2 =


0.025, which gives a 95% confidence interval. For tv=26a/2=002S = 2.056, y = 7.51 and sy = 0.266, the upper and lower 95% confidence limits are:


7.51 — 2.056(0.266) < n < 7.51 + 2.056(0.266) 6.96 < n < 8.05

= 8 mg/L


FIGURE 2.13 The t distribution for the estimated mean of the nitrate data with the 90% and 95% confidence intervals.

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