# Question 3:

## Part (a)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sample means: standard 107.2766, GM 113.6192 | B1 | |
| Sample variances: standard 138.2243, GM 121.9372 | B1 | |
| $H_0: \mu_1 = \mu_2$ (the means of the underlying distributions), $H_1: \mu_1 \neq \mu_2$ | B1 | Must be clear that hypotheses refer to underlying means |
| Test statistic: $\frac{113.6192 - 107.2766}{\sqrt{\frac{138.2243}{30} + \frac{121.93721}{26}}} = 2.08(0103)$ | M1A1 | |
| Compare with $z$ distribution | B1 | |
| Critical value, 2 tails, 5%, 1.960 | B1 | |
| So 2.08(0103) is just in the critical region | | |
| Hence reject $H_0$ and conclude there is a difference in mean weight between standard and GM tomatoes | B1B1 | |
| Assumption: the tomatoes may be regarded, in some sense, as random samples of their respective varieties | B1 [10] | |

## Part (a)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Additional assumptions: underlying Normality of the tomatoes' weights | B1 | |
| common variance in the underlying distributions | B1 | |
| Given the sample sizes it seems safe to use the Normal distribution | E1 | |
| The Normal test is better in that it makes fewer assumptions | E1 [4] | |

## Part (b)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| The tomatoes should be chosen at random | B1 | |
| The panel should not know which tomatoes are GM / standard | B1 | |
| Making fine judgements on the appearances of tomatoes is unlikely to be reliable | B1 [3] | Accept other sensible comments |

# Question 3 (continued):

## Part (ii) - Wilcoxon Rank Sum Test

| Answer | Marks | Guidance |
|--------|-------|----------|
| Rank sums: Standard 134, GM 76 | M1A1 | Or Mann-Whitney: 79, 21 |
| $H_0$: GM and std tomatoes have, on the whole, the same appearance | B1 | Hope for, but don't expect, a formulation in terms of a shift in location parameter for underlying distributions of appearances |
| $H_1$: GM tomatoes have, on the whole, a better appearance than std tomatoes | B1 | |
| (lower rank sum for GM indicates that the evidence is in the correct tail) | B1 | May be implied |
| Wilcoxon rank sum test | B1 | |
| Critical value for $m = n = 10$, 1 tail, 1% level is 74 | B1 | Critical value 19 |
| The observed value of 76 is not in the critical region | | |
| so accept $H_0$. That is, conclude that there is insufficient evidence to suppose that, on the whole, GM tomatoes have a better appearance than standard tomatoes. | B1 | |
| **Total** | **[7]** | |

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