# Question 5:

## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $B(n, 0.045)$ | B1 | For binomial with correct parameters $n$ and 0.045 |

## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| Applicants are independent (no identical twins) or the proportion/probability identified as colour blind does not change over time | B1 | For one of the given reasons. Must have context. Allow equivalent statements. Do not allow number for proportion/probability |

## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $B(120, 0.045) \Rightarrow \text{Po}(5.4)$ | B1 | Using or writing Po(5.4) |
| $P(X=5) = \frac{e^{-5.4} \times 5.4^5}{5!}$ | M1 | For $\frac{e^{-\lambda}\lambda^5}{5!}$ with any value of $\lambda$ |
| $= 0.1728\ldots$ awrt 0.173 | A1 | awrt 0.173. NB: A correct answer with no incorrect working scores 3/3 |

## Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| Binomial with large $n$ | B1 | $n$ is large (Allow "number of trials" for $n$) |
| and very small $p$ | B1 | |

## Part (e)
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: p = 0.75 \quad H_1: p \neq 0.75$ | B1 | |
| $B(96, 0.75) \Rightarrow N(72, 18)$ | B1 | |
| $Z = \frac{67.5 - 72}{\sqrt{18}}$ or $\frac{x \pm 0.5 - 72}{\sqrt{18}}$ | M1 | |
| $= -1.06066\ldots$ or $\frac{x+0.5-72}{\sqrt{18}} < -1.96$ or $\frac{x-0.5-72}{\sqrt{18}} > 1.96$ | A1 | |
| $P(z < -1.06) = 0.1444\ldots/0.1446$ or CR $< 63.2$ awrt 0.144 or 0.145 | A1 | |
| There is insufficient evidence to reject $H_0$ | dM1 | |
| Insufficient evidence against Jaymini's claim | A1 | |

**ALT (using probability of failing):**

| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: p = 0.25 \quad H_1: p \neq 0.25$ | B1 | |
| $B(96, 0.25) \Rightarrow N(24, 18)$ | B1 | |
| $Z = \frac{28.5-24}{\sqrt{18}}$ or $\frac{x \pm 0.5 - 24}{\sqrt{18}}$ | M1 | |
| $= 1.06066\ldots$ or $\frac{x+0.5-24}{\sqrt{18}} < -1.96$ or $\frac{x-0.5-24}{\sqrt{18}} > 1.96$ | A1 | |
| $P(z > 1.06) = 0.1444\ldots/0.1446$ or CR $> 32.8$ awrt 0.144 or 0.145 | A1 | |
| There is insufficient evidence to reject $H_0$ | dM1 | |
| Insufficient evidence against Jaymini's claim | A1 | |

# Question (e):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $p$ is small | B1 | Allow probability for $p$ |
| Both hypotheses correct in terms of $p$ or $\pi$, e.g. $H_0: p = $ value, $H_1: p \neq$ value | B1 | Must be attached to $H_0$ and $H_1$ |
| For writing or using $N(72, 18)$ | B1 | May be implied by a correct standardisation expression |
| Standardising using $67.5$ or $67$ or $66.5$ or $x \pm 0.5$ with their mean and standard deviation | M1 | Allow $\pm$ |
| awrt $-1.06$ (may be implied by awrt $0.144$ or $0.145$) or a correct standardisation with $\pm 1.96$ | A1 | Ignore incorrect inequality symbol and allow $=$ |
| Using probability route: awrt $0.144$ or $0.145$, or critical value $z = \pm 1.96$; Using critical region route: $CR < 63.2$ | A1 | |
| Correct statement – no context needed but do not allow contradicting non-contextual comments | dM1 | Dependent on M1 A1. Ignore any comparisons |
| Correct conclusion in context. Must include the word claim. If answer refers to the claim must include words applicants (oe) and pilots. No hypotheses then A0 | A1 | NB: Award M1 A1 for a correct contextual statement on its own |

**ALT:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| Both hypotheses correct in terms of $p$ or $\pi$ | B1 | Must be attached to $H_0$ and $H_1$ |
| For writing or using $N(24, 18)$ | B1 | May be implied by a correct standardisation expression |
| Standardising using $28.5$ or $29$ or $29.5$ or $x \pm 0.5$ with their mean and standard deviation | M1 | Allow $\pm$ |
| awrt $1.06$ (may be implied by awrt $0.144$ or $0.145$) or a correct standardisation with $\pm 1.96$ | A1 | Ignore incorrect inequality symbol and allow $=$ |
| Using probability route: awrt $0.144$ or $0.145$, or critical value $z = \pm 1.96$; Using critical region route: $CR < 32.8$ | A1 | |
| Correct statement – no context needed but do not allow contradicting non-contextual comments | dM1 | Dependent on M1 A1. Ignore any comparisons |
| Correct conclusion in context. Must include the word claim. Must include words applicants (oe) and pilots. No hypotheses then A0 | A1 | NB: Award M1 A1 for a correct contextual statement on its own |

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