# Question 5:

## Part (a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $H_0: p = 0.35$, $H_1: p > 0.35$ | B1 | Both hypotheses correct |
| $V \sim B(40, 0.35)$; $P(V \geq 18) = 1 - P(V \leq 17) = 0.1239$ or $P(V \geq 19) = 0.0699$, $P(V \geq 20) = 0.0363$; CR $V \geq 20$ | M1, A1 | awrt 0.124 or $V \geq 20$ or $V > 19$ |
| Accept $H_0$ / Not Significant / 18 does not lie in critical region | M1d | Dep on previous M; ft their CR or probability |
| There is insufficient evidence that the proportion/amount/number/percentage of customers who bought **organic vegetables** has increased | A1cso | Correct contextual statement; bold words required |

## Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $E \sim B(50, 0.35)$ | M1 | Or correct probability/one tail of CR correct |
| $P(E \leq 10) = 0.0160$; $P(E \geq 25) = 0.0207$; $P(E \leq 11) = 0.0342$; $P(E \geq 24) = 0.0396$ | | |
| CR: $E \leq 10$ and $E \geq 25$ | A1 A1 | Allow any letter; condone missing letter |

## Part (c):
| Working/Answer | Mark | Guidance |
|---|---|---|
| The **manager's claim** is supported or there is sufficient evidence that the proportion of customers buying organic **eggs** is different from those buying organic **vegetables** | B1ft | Must include: managers claim or eggs and vegetable(s); ft their 2-tail CR |

## Part (d):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $0.016 + 0.0207 = 0.0367$ or $3.67\%$ | B1 | awrt 0.0367 or 3.67% |

## Part (e):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $F \sim N(40, 32)$ | M1 A1 | Mean = 40, variance = 32 |
| $P(F < n) = P\!\left(Z < \frac{n - 0.5 - 40}{\sqrt{32}}\right)$ | M1 M1d | Continuity correction $n - 0.5$; dep on previous M |
| $\frac{n - 0.5 - 40}{\sqrt{32}} = -1.68$ | B1 | $\pm 1.68$ |
| $n = 31$ | A1cso | All previous marks must be awarded |

---