## 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 shorter for rats on a free-choice diet compared to rats on a healthy, restricted diet?

A study (Berger, Boos, and Guess 1988) compared the total lifetime of rats on restricted ($$n=106$$ rats) free-eating diets ($$n=89$$ rats).

1. Explain why these are two independent samples, and not paired.
2. Write down the hypotheses being tested. Is this a one- or a two-tailed test? Explain.
3. Explain what the ‘standard error of the difference’ would mean here.
4. What does the error bar chart in Fig. 9.1 tell us?
5. What are two possible reasons why the sample mean lifetimes of rats on the two diets are different?
6. Write down the $$t$$-score and the appropriate $$P$$-value, using the output in Fig. 9.2 (jamovi) or Fig. 9.3 (SPSS).
7. How are the differences defined? What do these mean? FIGURE 9.1: Error bar chart for the rat lifetime example FIGURE 9.2: The jamovi output summarising the rat lifetimes data FIGURE 9.3: The SPSS output summarising the rat lifetimes data

1. Make a conclusion, in context.
2. What conditions are necessary for the test to be statistically valid?
3. Is it reasonable to assume these conditions are satisfied? (You may, or may not, need to refer to Fig. 8.9.)
4. 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.