pre-menopausal is:

(0.13 − 1.96 × 0.033 to 0.13 + 1.96 × 0.033) = (0.065 to 0.195)

In other words you can be 95 per cent confident that the proportion of cases in this population who are pre-menopausal lies somewhere between 0.065 to 0.195. Or alternatively, that this interval represents a plausible range of values for the population proportion who are menopausal.

Exercise 9.4 Calculate the standard error for the sample proportion of controls in Table 1.6 who are pre-menopausal, and hence calculate the 95 per cent confidence interval for the corresponding population proportion. Interpret your result.

Estimating a confidence interval for the median of a single population

If your data is ordinal then the median rather than the mean is the appropriate measure of location (review Chapter 5 if you’re not sure why). Alternatively, if your data is metric but skewed (or your sample is too small to check the distributional shape), you might also prefer the median as a more representative measure. Either way a confidence interval will enable you to assess the likely range of values for the population median. As far as I know, SPSS does not calculate a confidence interval for a single median, but Minitab does, and bases its calculation on the Wilcoxon signed-rank test5 (I’ll discuss this in Chapter 12).

Table 9.2 Sample median pain levels, and 95 per cent confidence intervals for the difference between the two groups, at three time periods, in the analgesics/stump pain study. Reproduced courtesy of Elsevier (The Lancet, 1994, Vol No. 344, page 1724–6)

Pain assessed by visual analogue scale (0–100 mm).

5We won’t deal with tests (i.e. hypothesis tests) until we get to Chapter 12, but the confidence intervals that I discuss in this and in the next chapter are based on a number of different hypothesis tests. The alternative would have been for me to introduce hypothesis tests before I dealt with confidence intervals. However, for various pedagogic reasons I didn’t think this was appropriate.

An example from practice

Table 9.2 is from the analgesics and stump pain study referred to in Table 5.3, and shows the sample median pain levels and their 95 per cent confidence intervals (assessed using a visual analogue scale), for the treatment and control groups, at three time periods.

Exercise 9.5 In Table 9.2, interpret and compare the differences in median pain levels and their 95 per cent confidence intervals for each of the three time periods.