# Question 4:

## Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| $H_0$: No. of car breakdowns per month follows a Poisson distribution; $H_1$: No. of car breakdowns per month does not follow a Poisson distribution | B1 | Correct hypotheses mentioning "breakdowns" and "Poisson"; $Po(1.8)$ o.e. is B0 |

## Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| A Poisson distribution will assign some probability to (all) values greater than 5 (o.e.) | B1 | For a suitable reason mentioning the Poisson distribution taking values beyond 5, e.g. sample space for Poisson being $[0, \infty)$ |

## Part (c):
| Working | Mark | Guidance |
|---------|------|----------|
| Need $\hat{\lambda}$: $\hat{\lambda} = \frac{0\times12 + 1\times11 + 2\times19 + 3\times14 + 4\times3 + 5\times1}{(12+11+19+14+3+1)}$ or 1.8 | M1 | For an expression for the mean with at least 3 correct products and correct denominator; M0 if found $\lambda$ by working backwards from $P(X=0) \Rightarrow 60e^{-\lambda} = 9.92$ |
| [Under $H_0$ $X \sim Po(1.8)$] $E_1 = 60 \times P(X=1) = 17.85(227\ldots)$ | M1 | For a correct method for finding $E_1$ or $E_2$ (implied by one correct value) |
| $E_2 = 60 \times P(X=2) = 16.06(705\ldots)$ | A1 | For awrt 17.85 and awrt 16.07 |
| $E_{\geq 5} = 60 - \sum_{0}^{4} E_i = 2.18(43996\ldots)$ | B1ft | For awrt 2.18, or using $60 - (\text{sum of their } E_i)$ or $60 \times (1 - P(X \leq 4))$ |

## Part (d):
| Working | Mark | Guidance |
|---------|------|----------|
| $\frac{(11 - 17.85\ldots)^2}{17.85\ldots} = 2.6287\ldots$ or $\frac{(14 - 9.64)^2}{9.64} = 1.97195\ldots$ | M1 | For either correct expression (ft their $E_1$) |
| awrt 2.63 and awrt 1.97 | A1 | |

## Part (e):
| Working | Mark | Guidance |
|---------|------|----------|
| Need to combine last two columns since $E_i < 5$ | B1 | For explaining need to pool columns since $E_i < 5$; allow candidates describing combining last three columns as long as they say $E_i < 5$ |
| Degrees of freedom therefore $5 - 2$ since mean for Poisson estimated from $O_i$ | B1 | For mentioning mean/rate/parameter/$\lambda$ estimated from $O_i$ and 2 constraints |

## Part (f):
| Working | Mark | Guidance |
|---------|------|----------|
| $\chi_3^2(5\%) = 7.815$ | B1 | For correct cv of 7.815 (or better) |
| (Not significant) insufficient evidence to reject Anja's belief | B1 | For correct conclusion in context mentioning "breakdowns" and "Poisson"; allow equivalent words for belief such as "theory"; must be consistent with their cv; allow $Po(1.8)$ in conclusion instead of Poisson |