# Question 2:

## Part 2(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $z = 2.3263$ | B1 | 2.3263 or better |
| 98% CI for $\mu$: $128 \pm 2.3263 \times \frac{3.5}{\sqrt{80}}$ | M1 | For $128 \pm \sqrt{\frac{3.5^2}{80}} \times$ "z value", $|z|>1$. May be implied by correct CI |
| $= (127.0896..., 128.9103...) = \text{awrt } (127.09, 128.91)$ | A1 | awrt **127.09** and awrt **128.91** |

## Part 2(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \mu_{g(reen)} - \mu_{r(ed)} = 10$, $H_1: \mu_{g(reen)} - \mu_{r(ed)} > 10$ | B1 | Both hypotheses correct with correct notation (if using $\mu_x$ and $\mu_y$ these must be defined) |

## Part 2(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Normal model: $\bar{G} - \bar{R} \sim \text{N}\left(10, \frac{3.5^2}{80} + \frac{4^2}{n}\right)$ | M1 | Setting up normal model with correct mean or variance. May be shown in standardisation with correct mean or variance and $|z|>1$ |
| | A1 | Correct variance and mean, may be seen in standardisation |
| $z = \frac{(128-117)-10}{\sqrt{\frac{3.5^2}{80} + \frac{4^2}{n}}} > 1.6449$ | M1 | Setting up equation or inequality by standardising with their mean and standard deviation and $|z|>1$ |
| | B1 | Correct critical value 1.6449 or better, allow **1.64 or better** |
| $\frac{1}{1.6449} > \sqrt{\frac{3.5^2}{80} + \frac{4^2}{n}} \rightarrow \left(\frac{1}{1.6449}\right)^2 - \frac{12.25}{80} > \frac{16}{n} \rightarrow n > 73.9...$ | M1 | Rearranging to solve for $n$ (may be implied by 73.9...) |
| $n = 74$ | A1 | cao |

## Part 2(d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sample sizes are large so CLT means even though we do not know the distributions of $G$ and $R$, the means $\bar{G}$ and $\bar{G}-\bar{R}$ will be (approximately) normally distributed. | B1 | Correct explanation mentioning large sample size and the means being distributed normally |

## Part 2(e):
| Answer/Working | Mark | Guidance |
|---|---|---|
| [Since $85 > 73.9...$] there is significant evidence to support Camilo's belief/the mean weight of green apples is more than 10 grams greater than red apples is supported. | B1ft | Drawing a correct inference in context, must be consistent with their value of $n$ |

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