7.8 Optional: Confidence intervals for mean differences (paired data)

(Answers are available in Sect. A.7)

After a number of runners collapsed near the finish of the Tyneside annual Great North Run, researchers decided to study the role of \(\beta\) endorphins as a factor in the collapses. (\(\beta\) endorphins are a peptide which suppress pain.)

A study (Dale et al. 1987) examined what happened with plasma \(\beta\) endorphins during fun runs: how much do the concentrations change, on average?

The researchers recorded the plasma \(\beta\) concentrations in fun runners (participating in the Tyneside Great North Run). Eleven runners (who did not collapse) had their plasma \(\beta\) concentrations measured (in pmol/litre) before and after the race (Table 7.3.)

TABLE 7.3: Plasma-beta concentration for runners, before and after the race
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 \(\beta\) endorphines is usually less than about 11 pmol/litre.

  1. What do \(\mu_d\) and \(\bar{d}\) represent in this context?
  2. Explain why these data should be analysed as mean differences.
  3. Compute the changes in plasma \(\beta\) 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?)
  4. Using the statistics mode on your calculator, compute the sample mean difference \(\bar{d}\) and the sample standard deviation of the differences.
  5. Compute the standard error of the mean difference. Explain what this means in this context.
  6. 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?
  7. Compute an approximate 95% confidence interval for the population mean difference in plasma \(\beta\) concentrations.
  8. Construct a one-sentence statement that communicates a 95% CI for the population difference in plasma \(\beta\) concentrations.
  9. What conditions must be met for this CI to be statistically valid?
  10. Is it reasonable to assume the CI is statistically valid? Construct a stem-and-leaf plot to help.
  11. Do you think the population plasma \(\beta\) concentration changes during the race, on average? Explain.
  12. Suppose the researchers wished to estimate the change in plasma \(\beta\) endorphin concentrations to within \(2.5\) pmol/litre, with 95% confidence. What size sample would be necessary?

References

Dale, G., J. A. Fleetwood, Ann. Weddell, R. D. Ellis, and J. R. Sainsbury. 1987. “\(\beta\) Endorphin: A Factor in ‘Fun Run’ Collapse?” British Medical Journal 294: 1004.