In the hypothetical example on page 131 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 6.1 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 6.1 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:

and the 99% confidence limits are:

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.

FIGURE 6.6.

Visualizing the central limit theorem: The distribution of sample means approximates a normal distribution. Means for 10 (left) and 20 (right) samples of 5 from the 50 GDP values in table 6.1.

What Confidence Limits Are and What They Aren't

If you say that the 95% confidence limits for the estimated mean income are RM11,816-13,384, 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, SE, for each sample; and

3. If we then calculate the confidence intervals for each sample mean, based on ± 1.96 SE; then

4. 95% of these confidence intervals will contain the true mean, ^.