# Question 1:

## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $X \sim B(20, 0.05)$ or $Y \sim B(20, 0.95)$ | B1 | Writing or using $B(20,0.05)$. Allow $Y \sim B(20, 0.95)$ if $Y$ is clearly defined. Implied by 1 correct probability. |

## Part (a)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(X \leqslant 4) - P(X \leqslant 2)$ or $\binom{20}{3}0.05^3 \times 0.95^{17} + \binom{20}{4}0.05^4 \times 0.95^{16}$ | M1 | For $P(X \leqslant 4) - P(X \leqslant 2)$ and one correct prob, or $P(X=3) + P(X=4)$ and 1 correct prob |
| $= 0.072909...$ awrt $\mathbf{0.0729}$ | A1 | |

## Part (a)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(X \leqslant 1)$ or $P(Y \geqslant 19) = 20 \times (0.95)^{19}(0.05) + (0.95)^{20}$ | M1 | For $P(X \leqslant 1)$ or $[20]\times(0.95)^{19}(0.05)+(0.95)^{20}$ – condone missing 20 |
| $= 0.7358$ awrt $\mathbf{0.736}$ | A1 | (5 marks total for (i) and (ii)) |

## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Let $W$ = no. of packets where $Y > 18$; $P(W=5) = (0.7358...)^5$ | M1 | For $(\text{their}(a)(ii))^5$ |
| $= 0.21573...$ awrt $\mathbf{0.216}$ | A1 | (2 marks) |

## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: p = 0.05 \quad H_1: p > 0.05$ | B1 | Both hypotheses correct with $p$ or $\pi$ (1 mark) |

## Part (d)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $V \sim B(100, 0.05)$ approximated by $V \sim \text{Po}(5)$ | M1A1 | 1st M1: realising Poisson approximation appropriate. NB Po(95) is M0A0. 1st A1: correct mean for Poisson |
| $P(V \geqslant 8) = 1 - P(V \leqslant 7) = 1 - 0.8666 = 0.1334$ | M1 | 2nd M1: for $1 - P(V \leqslant 7)$ or $P(V \leqslant 7) = 0.8666$; or writing $P(V \geqslant 10) = 0.0318$ or $P(V \geqslant 9) = 0.0681$ or $P(V \geqslant 11) = 0.0137$ leading to a CR |
| CR for 1-tail: $V \geqslant 10$ oe; CR for 2-tail: $V \geqslant 11$ oe | A1 | 2nd A1: awrt 0.133 or $V \geqslant 10$ oe (e.g. $V > 9$) or $V \geqslant 11$ oe; allow any letter but CR must match part (c) |
| Accept $H_0$ or not significant or 8 does not lie in the critical region | dM1 | 3rd dM1: dep on 2nd M1; ft their CR or probability. Correct statement comparing 8 with their CR or their prob with 0.05 or 0.025 |
| Data consistent with **Spany's claim** or Insufficient evidence for **Jem's belief** or insufficient evidence that **percentage of seeds not germinating** is more than **5%** | A1cso | All previous marks must be awarded. **Correct** statement in context. Need **Bold words**. NB award M1A1 for correct contextual statement on its own. (6 marks) |

**Total: 14 marks**

### Special Cases:
| SC | Guidance |
|---|---|
| SC1 | Sight of $N(5$ or $95,\ \sqrt{4.75}^2)$ M1A1; probability awrt 0.125/6 M1A1; Correct contextual conclusion dM1A1 |
| SC2 | No approximation: Use of $B(100, 0.05)$ M0A0; probability awrt 0.128 or $CR \geqslant 10$ M1A1; then M0A0 |