# Statistics for Environmental Engineers

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Halving the standard error is a big gain. If the true difference between two treatments is one standard error, there is only about a 17% chance that it will be detected at a confidence level of 95%. If the true difference is two standard errors, there is slightly better than a 50/50 chance that it will be identified as statistically significant at the 95% confidence level.

We now see the dilemma for the engineer and the statistical consultant. The engineer wants to detect a small difference without doing many replicates. The statistician, not being a magician, is constrained to certain mathematical realities. The consultant will be most helpful at the planning stages of an experiment when replication, randomization, blocking, and experimental design (factorial, paired test, etc.) can be integrated.

What follows are recipes for a few simple situations in single-factor experiments. The theory has been mostly covered in previous chapters.

### Confidence Interval for a Mean

The (1 — a)100% confidence interval for the mean n has the form y ± E, where E is the half-length E = za/2a/jn. The sample size n that will produce this interval half-length is:

za/2a^

The value obtained is rounded to the next highest integer. This assumes random sampling. It also assumes that n is large enough that the normal distribution can be used to define the confidence interval. (For smaller sample sizes, the t distribution is used.)

To use this equation we must specify E, a or 1 — a, and a. Values of 1 — a that might be used are:

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