## 10.2 Hypothesis tests for two means

(This continues from Sect. 9.2.)

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 et al. 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. 10.1 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. 10.2 (jamovi) or Fig. 10.3 (SPSS).
- Calculate the \(t\)-score (using the standard error as given in the output), and show it is the same value as given in the output.
- 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. 9.5.)
- What if all these rats only came from only 20 litters?

### References

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