## 8.3 CIs for difference between two sample means

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

For rats, is the mean lifetime shorter for rats on a free-choice diet compared to rats on a healthy, restricted diet?

A study 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 (Fig. 8.1).

1. Explain why this study compares two independent groups.
2. What is the parameter of interest?
3. Use the numerical summary (Fig. 8.2 (jamovi); Fig. 8.3 (SPSS)) and the output from the analysis (Fig. 8.2 (jamovi); Fig. 8.4 (SPSS)) to write down the appropriate $$95$$% CI, to estimate the difference between the population means.
1. Which of these short statements best communicates this CI? Why are the other statements incorrect?

1. The sample mean lifetime is between $$223.34$$ and $$346.13$$ days.
2. The difference between the population mean lifetimes is between $$223.34$$ and $$346.13$$ days.
3. We are $$95$$% sure that the difference between the sample mean lifetimes is between $$223.34$$ and $$346.13$$ days.
4. We are $$95$$% sure that the difference between the population mean lifetimes is between $$223.34$$ and $$346.13$$ days.
5. If we repeated everything many times, $$95$$% of the CIs constructed would contain the difference between the population means.
1. The above statement communicating the CI isn't perfect. Why not? Write an improved statement communicating the CI.
2. 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. 8.5.)
3. Explain the difference between the meaning of what is displayed in the two graphs in Fig. 8.5.

### References

Berger RL, Boos DD, Guess FM. Tests and confidence sets for comparing two mean residual life functions. Biometrics. 1988;44(1):103–15.