# Question 3:

## Part 3(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\text{apple} \geq 220\text{g}) = P\!\left(Z > \frac{220 - 210.5}{15.2}\right)$ | M1 | For standardising; condone numerator reversed but penalise continuity corrections |
| $= P(Z > 0.625) = 1 - \Phi(0.625) = 1 - 0.7340$ | M1 | For correct structure |
| $= 0.2660$ | A1 CAO | i.e. $1 - \Phi(\text{positive } z \text{ value})$; allow 0.266 but not 0.27 |
| **Total: 3** | | |

## Part 3(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\text{all 3 weigh} \geq 220\text{g}) = 0.2660^3 = 0.0188$ | M1 | M1 for their answer to part (i) cubed |
| | A1 FT | Allow 0.019 and 0.01882 |
| **Total: 2** | | |

## Part 3(iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| (A) Binomial $(100, 0.0188)$ | B1 | For binomial |
| | B1 | For parameters; second B1 dependent on first B1; FT their answer to part (ii) for second B1 |
| (B) Use Poisson with $\lambda = 100 \times 0.0188 = 1.88$ | B1 | For Poisson SOI; consistent with $p < 0.1$ from part (iii)(A) |
| $P(\text{at most one}) = e^{-1.88}\frac{1.88^0}{0!} + e^{-1.88}\frac{1.88^1}{1!}$ | B1 | For Poisson mean; FT answer to part (ii) with $p < 0.1$ |
| $= 0.1525 + 0.2869 = 0.4394$ | M1 | For either probability |
| | M1 | For sum of both |
| | A1 CAO | For 0.44 or better; allow 0.4395 but not 0.4337 |
| (C) $n$ is large and $p$ is small | B1 | Allow '$n$ is large and $np < 10$' and '$n$ is large and $np \approx npq$' |
| **Total: 5+1** | | |

## Part 3(iv)(A)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\Phi^{-1}(0.1) = -1.282$ | B1 | For $\pm 1.282$ |
| $\frac{170 - 185}{\sigma} = -1.282$ | M1 | For correct equation as written o.e.; allow M1 if different $z$-value used |
| $1.282\sigma = 15 \Rightarrow \sigma = 11.70$ | A1 CAO | Without incorrect working seen; allow 11.7 |
| **Total: 3** | | |

## Part 3(iv)(B)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Two Normal curves drawn, intersecting | G1 | For shape; ignore labelling of vertical axis |
| Means shown at 185 and 210.5 | G1 | For means shown explicitly or by scale |
| Cox's curve has lower maximum height | G1 | For lower max height for Braeburns |
| Braeburns curve has greater width | G1 | For greater width (variance) for Braeburns |
| **Total: 4** | | |

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