Statistics Review - Mathematics and Statistics for the Environment - page 122

7.11.1.3 Confidence Limits
Sample estimates are subject to variation. How much they vary depends primarily on the inherent
variability of the population ( Var 2 ) and on the size of the sample ( n ) and of the population ( N ). The
statistical way of indicating the reliability of an estimate is to establish confidence limits. For esti-
mates made from normally distributed populations, the confidence limits are given by

Estimate ± t (standard error)

For setting confidence limits on the mean and total we already have everything we need except for
the value of t , and that can be obtained from a table of the t distributi on .
In the previous example, the sample of n = 20 units had a mean of X = 10.7 and a standard error
of s x = 0.734. For 95% confidence limits on the mean we would use a t value (from a t table) of 0.05
and (also from a t table) 19 degrees of freedom. As t 0.05 = 2.093, the confidence limits are given by

Xt ± ()() s x = 10 72093 . ± ( . )(. 0 734 ) = 9.16 to 12.24

This says that, unless a 1-in-20 chance has occurred in sampling, the population mean is somewhere
between 9.16 and 12.24. It does not say where the mean of future samples from this population
might fall, nor does it say where the mean may be if mistakes have been made in the measurements.
For 99% confidence limits, we find t 0.01 = 2.861 (with 19 degrees of freedom), so the limits are

10.7 ± (2.861)(0.734) = 8.6 to 12.8

These limits are wider, but they are more likely to include the true population mean. For the popula-
tion total the confidence limits are

95% limits = 1070 ± (2.093)(73.4) = 916 to 1224

99% limits = 1070 ± (2.861)(73.4) = 860 to 1280

For large samples ( n > 60), the 95% limits are closely approximated by

Estimate ± 2 (standard error)

and the 99% limits by

Estimate ± 2.6 (standard error)

7.11.1.4 Sample Size
Samples cost money. So do errors. The aim in planning a survey should be to take enough obser-
vations to obtain the desired precision—no more, no less. The number of observations needed in
a simple random sample will depend on the precision desired and the inherent variability of the
population being sampled. Because sampling precision is often expressed in terms of the confidence
interval on the mean, it is not unreasonable in planning a survey to say that in the computed confi-
dence interval

X s x ±

we would like to have the ts x equal to or less than some specified value E , unless a 1-in-20 (or 1-in-
100) chance has occurred in the sample. That is, we want

ts x = E