## 9.1 Hypothesis tests for two means

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 total lifetime of rats on restricted (\(n=106\) rats) free-eating diets (\(n=89\) rats).

- Explain why these are two
*independent*samples, and not*paired*. - Write down the hypotheses being tested. Is this a one- or a two-tailed test? Explain.
- Explain what the ‘standard error of the difference’ would mean here.
- What does the error bar chart in Fig. 9.1 tell us?
- What are
*two*possible reasons why the sample mean lifetimes of rats on the two diets are different? - Write down the \(t\)-score and the appropriate \(P\)-value, using the output in Fig. 9.2 (jamovi) or Fig. 9.3 (SPSS).
- How are the differences defined? What do these mean?

- Make a conclusion, in context.
- What conditions are necessary for the test to be statistically valid?
- Is it reasonable to assume these conditions are satisfied? (You may, or may not, need to refer to Fig. 8.9.)
- What if all these rats only came from only 20 litters?

### 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.