S3 January 2006 Q6
6. An area of grass was sampled by placing a 1 m × 1 m square randomly in 100 places. The numbers of daisies in each of the squares were counted. It was decided that the resulting data could be modelled by a Poisson distribution with mean 2. The expected frequencies were calculated using the model.
The following table shows the observed and expected frequencies.
| Number of daisies | Observed frequency | Expected frequency |
|---|---|---|
| 0 | 8 | 13.53 |
| 1 | 32 | 27.07 |
| 2 | 27 | \(r\) |
| 3 | 18 | \(s\) |
| 4 | 10 | 9.02 |
| 5 | 3 | 3.61 |
| 6 | 1 | 1.20 |
| 7 | 0 | 0.34 |
| \(\geqslant 8\) | 1 | \(t\) |
(a) Find values for \(r\), \(s\) and \(t\). (4)
(b) Using a 5% significance level, test whether or not this Poisson model is suitable. State your hypotheses clearly. (7)
An alternative test might have been to estimate the population mean by using the data given.
(c) Explain how this would have affected the test. (2)
| Scheme | Marks |
|---|---|
| \(r = 27.07\), | M1 A1 |
| \(s = 18.04\), | B1 |
| \(t = 0.11\) using tables or 0.12 using totals | B1ft |
| (4) |
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0\) : A Poisson model Po(2) is a suitable model. \(\mathrm{H}_1\) : A Poisson model Po(2) is not a suitable model. both | B1 |
| Amalgamate data | M1 |
| \(\displaystyle\sum \frac{(O - E)^2}{E} = 3.28\) (awrt) | M1 A1 |
| \(\nu = 6 - 1 = 5\) | B1 |
| \(\chi^2_5(5\%) = 11.070\) (follow through their degrees of freedom) | B1ft |
| \(3.28 \lt 11.070\) There is insufficient evidence to reject \(\mathrm{H}_0\), Po(2) is a suitable model. | A1ft |
| (7) |
Notes
(corrected from the printed mark scheme: the comparison is printed as “3.25 < 11.070”; the test statistic found above is 3.28)
| Scheme | Marks |
|---|---|
| The expected values, and hence \(\sum \dfrac{(O - E)^2}{E}\) would be different, and the degrees of freedom would be 1 less. | B1 B1 |
| (2) | |
| (13 marks) |