# Statistics for Environmental Engineers

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-t    < ibn < t

tv,a/2 —    rlv,al2

s/Jn

will be satisfied- This means that the maximum value of the error |y — П is:

ly — П = rv,a/2 r

n

with probability 1 — a— In other words, we can assert with probability 1 — a that the error in using y to estimate n will be at most tv a/2 ——-

From here, the comparison of the estimated mean with the standard value can be done as a hypothesis test or by computing a confidence interval- The two approaches are equivalent and will lead to the same conclusion- The confidence interval approach is more direct and often appeals to engineers-

### Testing the Null Hypothesis

The comparison between y and n0 can be stated as a null hypothesis:

Hо: n — По = 0

which is read “the expected difference between n and По is zero-” The “null” is the zero- The extent to which y differs from n will be due to only random measurement error and not to bias- The extent to which y differs from По will be due to both random error and bias- We hypothesize the bias (n По) to be zero, and test for evidence to the contrary-The sample average is:

n

The sample variance is:

s2 = Ky-y-

s = ——n———1——-

and the standard error of the mean is: