7.3 Confidence intervals for mean differences (paired data)

A study (Guirao et al. 2017) examined the difference between 2-minute walk times (2MWT) for 10 patients before and after receiving a prosthetic implant. (The 2MWT measures how far participants can walk in two minutes.)

The 2MWT for ten amputees with and without an implant are shown in Fig. 7.4.

Two-minute Walk Times (2MWT) for 10 patients with implants (With Imp) and without implants (Without Imp), in metres

FIGURE 7.4: Two-minute Walk Times (2MWT) for 10 patients with implants (With Imp) and without implants (Without Imp), in metres

  1. What type of RQ is implied: Descriptive, relational, or interventional?
  2. What do \(\mu_d\) and \(\bar{d}\) represent in this context?
  3. Explain why these data should be analysed as paired data.
  4. Compute the changes in 2MWT for each patient.
  5. Although it doesn’t really matter, why does it probably makes more sense to compute the With Imp values minus the Without Imp values?
  6. Using the statistics mode on your calculator, compute the sample mean difference \(\bar{d}\) and the sample standard deviation of the differences \(s_d\).
  7. Compute the standard error of the mean difference \(\text{s.e.}(\bar{d})\). Explain what this tells us in this context.
  8. If another sample of ten subjects were studied, would the same sample mean difference \(\bar{d}\) be computed? How much variation would be expected in the sample mean differences found from different samples?
  9. Draw the approximate sampling distribution of \(\bar{d}\).
  10. Compute an approximate 95% confidence interval for the population mean difference in 2MWT.
  11. Construct a one-sentence statement that communicates a 95% CI for the population change in 2MWT.
  12. What conditions must be met for this CI to be statistically valid?
  13. Is it reasonable to assume the CI is statistically valid? Construct a stem-and-leaf plot to help.
  14. Suppose the researchers wished to estimate the change in 2MWT to within \(5\)m, with 95% confidence. What size sample would be necessary?
  15. Do you think the population 2MWT changes because of the prosthetic, on average? Explain and discuss with your group.


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References

Guirao, Lluis, C. Beatriz Samitier, Maria Costea, Josep Maria Camos, Maria Majo, and Eulogio Pleguezuelos. 2017. “Improvement in Walking Abilities in Transfemoral Amputees with a Distal Weight Bearing Implant.” Prosthetics and Orthotics International 4 (26–32).