# Question 3:

## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \mu_A = \mu_B$ | 1 | Do NOT allow $\bar{X} = \bar{Y}$ or similar |
| $H_1: \mu_A \neq \mu_B$ | 1 | Accept absence of "population" if correct notation $\mu$ is used. Hypotheses stated verbally must include the word "population" |
| Test statistic: $\dfrac{26.4 - 25.38}{\sqrt{\dfrac{2.45}{7} + \dfrac{1.40}{5}}}$ | M1 | Numerator |
| | M1 | Denominator two separate terms correct |
| | M1 | |
| $\dfrac{1.02}{\sqrt{0.63}} = \dfrac{1.02}{0.7937} = 1.285$ | A1 | |
| Refer to $N(0,1)$ | 1 | No FT if wrong |
| Double-tailed 5% point is 1.96 | 1 | No FT if wrong |
| Not significant | 1 | |
| No evidence that the population means differ | 1 | |

## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| CI for $(\mu_A - \mu_B)$ is $1.02 \pm$ | M1 | |
| $1.645 \times$ | B1 | |
| $0.7937 =$ | M1 | |
| $1.02 \pm 1.3056 = (-0.2856, 2.3256)$ | A1 cao | Zero out of 4 if not $N(0,1)$ |

## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$ is accepted if $-1.96 <$ test statistic $< 1.96$ | M1 | SC1 Same wrong test can get M1,M1,A0 |
| i.e. if $-1.96 < \dfrac{\bar{x}-\bar{y}}{0.7937} < 1.96$ | M1 | SC2 Use of 1.645 gets 2 out of 3 |
| i.e. if $-1.556 < \bar{x} - \bar{y} < 1.556$ | A1 | BEWARE PRINTED ANSWER |
| In fact, $\bar{X} - \bar{Y} \sim N(2, 0.7937^2)$ | M1 | |
| So we want $P(-1.556 < N(2, 0.7937^2) < 1.556)$ | M1 | |
| $P\left(\dfrac{-1.556-2}{0.7937} < N(0,1) < \dfrac{1.556-2}{0.7937}\right)$ | M1 | Standardising |
| $P(-4.48 < N(0,1) < -0.5594) = 0.2879$ | A1 cao | |

## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Wilcoxon would give protection if assumption of Normality is wrong | E1 | |
| Wilcoxon could not really be applied if underlying variances are indeed different | E1 | |
| Wilcoxon would be less powerful (worse Type II error behaviour) with such small samples if Normality is correct | E1 | |

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