## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $Y_{ij}=\mu+\alpha_i+\varepsilon_{ij}$ | B1 | |
| where $Y_{ij}$ is the $j$th value in the $i$th group | B1 | |
| $\mu$ is the global mean in the underlying population | B1 | |
| $\alpha_i$ is the 'treatment effect' in the $i$th group | B1 | Or $\mu_i-\mu$ |
| $\varepsilon_{ij}$ is a random error term | B1 | Accept "residual" |
| In this context, $\mu$ measures the average effect of the exercise regimes, and the $\alpha_i$ represent the differences from the mean for the three regimes | E1 | Context explained at least once; 'Groups' are exercise regimes |
| $\varepsilon_{ij}$ iid $N(0,\sigma^2)$ | B1 | Distributional assumption |
| $H_0$: the three exercise regimes give the same (population) increase in mean pulse rate | B1 | Or: $\alpha_1=\alpha_2=\alpha_3(=0)$ |
| $H_1$: the three exercise regimes do not give the same (population) increase in mean pulse rate | | Not all $\alpha_i$ the same |
| $\sum\frac{T_i^2}{n_i}-\frac{T^2}{n}=\frac{244^2}{5}+\frac{368^2}{6}+\frac{266^2}{5}-\frac{878^2}{16}=448.8167$ | M1A1 | |
| $\sum\sum y_{ij}^2-\frac{T^2}{n}=49544-\frac{878^2}{16}=1363.75$ | M1A1 | |
| ANOVA table: Between Groups SS = 448.8167, df = 2, MS = 224.41, F ratio = 3.1885, F critical = 3.8056 | A1Ft, B1 | Within Groups Sum Sq; Df all 3 |
| Within Groups SS = 914.9333, df = 13, MS = 70.379 | A1Ft | F ratio Ft their Sum Sqs; Ft their Total SS-BGSS |
| Total SS = 1363.75, df = 15 | B1 | F critical; 3.81 from tables |
| Result not significant | M1 | |
| Insufficient evidence to suppose that the exercise regimes have different effects on pulse rate on average | A1 | |

## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| The analysis using three tests is not equivalent to ANOVA, and the multiple comparisons procedure is worse than ANOVA | B1 | Other points could be made; e.g. Multiple comparisons are likely to generate more type I errors than the nominal significance level would suggest |
| The three tests are not independent | B1 | |
| The significance level of the whole procedure is therefore impossible to assess | B1 | However, multiple comparisons are useful post hoc to identify where the largest differences have occurred |
| A comparison with the different result obtained in (i) | B1 | |
| and why this may be so | B1 | |