## Question 8:

Find mean of sample data [for use in Poisson distribution]:

$\lambda = 220/100 = 2.2$ | B1 |

State null hypothesis:

$H_0$: Poisson distribution fits data *or* $\lambda = 2.2$ | B1 |

Find expected values $\frac{100\lambda^r e^{-\lambda}}{r!}$ (to 1 d.p.):

$11.080\quad 24.377\quad 26.814\quad 19.664\quad 10.8151\quad 4.759\quad 2.491$ | M1 A1 | (ignore incorrect final value here for M1)

Combine last two cells so that exp. value $\geq 5$:

$O_i: 3$

$E_i: 7.25$ | M1* |

Calculate value of $\chi^2$ (to 2 d.p.; A1 dep M1*):

$\chi^2 = 0.076 + 2.879 + 0.653 + 1.448 + 0.441 + 2.491 = 7.99$ | M1 A1 | (allow 7.95 if 1 d.p. exp. values used)

State or use consistent tabular value (to 3 s.f.):

5 cells: $\chi^2_{3,\,0.95} = 7.815$

6 cells: $\chi^2_{4,\,0.95} = 9.488$ (correct)

7 cells: $\chi^2_{5,\,0.95} = 11.07$ | B1 |

State or imply valid method for conclusion:

Accept $H_0$ if $\chi^2 <$ tabular value | M1 |

Conclusion (AEF, requires both values correct):

Distribution fits *or* $\lambda = 2.2$ | DA1 | Not combining cells [so $\chi^2 = 8.64$] can earn B1 B1 M1 A1 M0 M1 B1 M1 (max 7)

**Total: 10 marks**