Now we are really close to answering the question about sample size. Suppose we want to get the standard error down to RM200 instead of RM400. We need to solve the following equation:

Solving for n:

In other words, to reduce the standard error of the mean from RM400 to RM200, we have to increase the sample size from 100 to 400 people.

Suppose we increase the sample to 400 and we still get a mean of RM12,600 and a standard deviation of RM4000. The standard error of the mean would then be RM200, and we could estimate, with 95% confidence, that the true mean of the population was between RM12,208 and 12,992. With just 100 people in the sample, the 95% confidence limits were RM11,816-13,384. As the standard error goes down, we get narrower—that is, more precise—confidence limits.

Let’s carry this forward another step. If we wanted to get the standard error down to RM100 and the 95% confidence interval down to RM200 from RM400, we would need a sample of 1,600 people. There is a pattern here. To cut the 95% confidence interval in half, from RM800 to RM400, we had to quadruple the sample size from 100 to 400. To cut the interval in half again, to RM200, we’d need to quadruple the sample size again, from 400 to 1,600.

There is another pattern, too. If we want to increase our confidence from 95% to 99% that the true mean of the population is within a particular confidence interval, we can raise the multiplier in formula 6.2 from roughly 2 standard deviations to roughly 3. Using the confidence interval of RM400, we would calculate:

We need 900 people, not 400, to be about 99% confident that our sample mean is within RM400, plus or minus, of the parameter.