Russel-Research-Method-in-Anthropology

178 Chapter 7
dard deviation is down to 16.12, and the mean is $291.54—very close to the
parameter of $294.16.
We are much closer to answering the question: How big does a sample have
to be?
The Standard Error and Confidence Intervals
In the hypothetical example on page 170 we took a sample of 100 merchants
in a Malaysian town and found that the mean income was RM12,600,
standard error 400. We know from figure 7.1 that 68.26% of all samples of
size 100 from this population will produce an estimate that is between 1 standard
error above and 1 standard error below the mean—that is, between
RM12,200 and RM13,000. The 68.26% confidence interval, then, is $400.
We also know from figure 7.1 that 95.44% of all samples of size 100 will
produce an estimate of 2 standard errors, or between RM13,400 and
RM11,800. The 95.44% confidence interval, then, is RM800. If we do the
sums for the example, we see that the 95% confidence limits are:
RM12,600 1.96(RM400) RM11,816 to RM13,384
and the 99% confidence limits are:
RM12,600 2.58(RM400) RM11,568 to RM13,632
Our ‘‘confidence’’ in these 95% or 99% estimates comes from the power of
a random sample and the fact that (by the central limit theorem) sampling
distributions are known to be normal irrespective of the distribution of the
variable whose mean we are estimating.
What Confidence Limits Are and What They Aren’t
If you say that the 95% confidence limits for the estimated mean income
are RM11,816 to RM13,384, this does not mean that there is a 95% chance
that the true mean, , lies somewhere in that range. The true mean may or
may not lie within that range and we have no way to tell. What we can say,
however, is that:
1. If we take a very large number of suitably large random samples from the population
(we’ll get to what ‘‘suitably large’’ means in a minute); and
2. If we calculate the mean, x, and the standard error, SEM, for each sample; and
3. If we then calculate the confidence intervals for each sample mean, based on
1.96 SEM; then
4. 95% of these confidence intervals will contain the true mean, μ