## Question 7:

### Part (i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \lambda = 5$ (1 or $\frac{1}{3}$), $H_1: \lambda < 5$ (1 or $\frac{1}{3}$) | B2 | One error, B1, except $t$, $x$ etc: 0. Allow $\mu$ |
| $\text{Po}(5)$ | M1 | Stated or implied |
| $P(\leq 2) = \mathbf{0.1247}$ | A1 | |
| $> 0.05$ | A1 | |
| $\beta$: CR is $\leq 1$ and compare 2 explicitly, $p = 0.0404$ | A1*, dep*A1 | |
| Do not reject $H_0$. Insufficient evidence of reduction in mean number of points failures | M1, A1ft | Correct first conclusion, needs $\text{Po}(5)$. Contextualised, acknowledge uncertainty. FT on wrong CR/$p$/comparison value |
| **Total: 7** | | |

### Part (ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Po}(50) \approx N(50, 50)$ | M1, M1 | Stated or implied. Normal, mean $150\lambda$ |
| | A1 | Variance/sd same |
| $50 - 0.5 - 1.645\times\sqrt{50}$ | M1 | $50 - z\times\sqrt{50}$, *not* $\div\sqrt{n}$, allow $\sqrt{}$ error, any cc |
| $= 37.87$ | A1 | $z = 1.645$, need $\sqrt{50}$, allow $50 + 0.5$ or 50 |
| so maximum number is **37** | A1 | 37 only, from cc, *not* final answer 37.86, but allow corrected to 37 after check |
| **Total: 6** | | 38(.36) from no cc is probably 5/6. Exact Poisson $(0.0473 \Rightarrow 38)$ is 1/6 |

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