7.10 Optional: CI for mean differences (paired data)
This question is optional; e.g., if you need more practice, or you are studying for the exam.
(Answers are available in Sect. A.7)
This question has a video solution in the online book, so you can hear and see the solution.
After a number of runners collapsed near the finish of the Tyneside annual Great North Run, researchers decided to study the role of β endorphins as a factor in the collapses. (β endorphins are a peptide which suppress pain.)
A study (Dale et al. 1987) examined what happened with plasma β endorphins during fun runs: how much do the concentrations change, on average?
The researchers recorded the plasma β concentrations in fun runners (participating in the Tyneside Great North Run). Eleven runners (who did not collapse) had their plasma β concentrations measured (in pmol/litre) before and after the race (Table 7.3.)
| Before race | After race | Difference |
|---|---|---|
| 4.3 | 29.6 | 25.3 |
| 4.6 | 25.1 | 20.5 |
| 5.2 | 15.5 | 10.3 |
| 5.2 | 29.6 | 24.4 |
| 6.6 | 24.1 | 17.5 |
| 7.2 | 37.8 | 30.6 |
| 8.4 | 20.1 | 11.7 |
| 9.0 | 21.9 | 12.9 |
| 10.4 | 14.2 | 3.8 |
| 14.0 | 34.6 | 20.6 |
| 17.8 | 46.2 | 28.4 |
The 'usual' value for β endorphines is usually less than about 11 pmol/litre.
- What do μd and ˉd represent in this context?
- Explain why these data should be analysed as mean differences.
- Compute the changes in plasma β endorphin concentration during the run (i.e. the 'differences'). (Although it doesn't really matter, why does it probably makes more sense to compute the after values minus the before values?)
- Using the statistics mode on your calculator, compute the sample mean difference ˉd and the sample standard deviation of the differences.
- Compute the standard error of the mean difference. Explain what this means in this context.
- If another sample of eleven runners were studied, would the same sample mean difference be computed? How much variation would be expected in the sample means found from different samples?
- Compute an approximate 95% confidence interval for the population mean difference in plasma β concentrations.
- Construct a one-sentence statement that communicates a 95% CI for the population difference in plasma β concentrations.
- What conditions must be met for this CI to be statistically valid?
- Is it reasonable to assume the CI is statistically valid? Construct a stem-and-leaf plot to help.
- Do you think the population plasma β concentration changes during the race, on average? Explain.
- Suppose the researchers wished to estimate the change in plasma β endorphin concentrations to within 2.5 pmol/litre, with 95% confidence. What size sample would be necessary?