## 8.2 Confidence intervals for difference between two independent samples

Researchers were interested in the impact of diet on the lifetime of rats:

For rats, is the mean lifetime

shorterfor rats on afree-choicediet compared to rats on ahealthy, restricteddiet?

A study (Berger, Boos, and Guess 1988) compared the lifetime of rats on a healthy, restricted diet (106 rats) and on a free-eating diet (89 rats).

The data set is large, so we only show an extract of the data (in jamovi: Fig. 8.4; in SPSS: Fig. 8.5).

- Explain why this study compares two
*independent*samples. - What is the
**parameter**of interest? - Use the numerical summary
(Fig. 8.6 (jamovi); Fig. 8.7 (SPSS))
and the output from the analysis
(Fig. 8.6 (jamovi); Fig. 8.8 (SPSS))
to write down the appropriate 95% CI,
to estimate
the
*difference*between the population means.

Which of these short statements best communicates this CI?

**Why**are the other statements incorrect?- The sample mean lifetime is between \(223.34\) and \(346.13\) days.
- The population mean difference in lifetimes is between \(223.34\) and \(346.13\) days.
- We are 95% sure that the sample mean difference in lifetimes is between \(223.34\) and \(346.13\) days.
- We are 95% sure that the difference between the population mean lifetimes is between \(223.34\) and \(346.13\) days.
- If we repeated everything many times, 95% of the CIs constructed would contain the difference between the population mean.

- The above statement communicating the CI isn’t perfect. Why not? Write an improved statement communicating the CI.
- What conditions must be met for this CI to be statistically valid? Is it reasonable to assume the CI is statistically valid? (You may, or may not, need to refer to Fig. 8.9.)
- Explain the difference between the
*meaning*of what is displayed in the two graphs shown in Fig. 8.10.

### References

Berger, Roger L., Dennis D. Boos, and Frank M. Guess. 1988. “Tests and Confidence Sets for Comparing Two Mean Residual Life Functions.” *Biometrics* 44 (1): 103–15.