S3 June 2010 Q6
6. A total of 228 items are collected from an archaeological site. The distance from the centre of the site is recorded for each item. The results are summarised in the table below.
| Distance from the centre of the site (m) | 0–1 | 1–2 | 2–4 | 4–6 | 6–9 | 9–12 |
| Number of items | 22 | 15 | 44 | 37 | 52 | 58 |
Test, at the 5% level of significance, whether or not the data can be modelled by a continuous uniform distribution. State your hypotheses clearly. (12)
| Scheme | Marks | |||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| M1 | |||||||||||||||||||||||||||||||||||
| A1 | |||||||||||||||||||||||||||||||||||
| A1 | |||||||||||||||||||||||||||||||||||
| M1 A1 | |||||||||||||||||||||||||||||||||||
| \(\mathrm{H}_0\): continuous uniform distribution is a good fit \(\mathrm{H}_1\): continuous uniform distribution is not a good fit | B1 | |||||||||||||||||||||||||||||||||||
| \(\displaystyle\sum\frac{(O_i - E_i)^2}{E_i} = \frac{313}{114} = 2.75\) or \(\displaystyle\sum\frac{O_i^2}{E_i} - 228 = 230.745\ldots - 228 = \ldots\) (awrt 2.75) | dM1A1 | |||||||||||||||||||||||||||||||||||
| \(\nu = 6 - 1 = 5\) | B1 | |||||||||||||||||||||||||||||||||||
| \(\chi^2_5(0.05) = 11.070\) (ft their \(\nu\) i.e. \(\chi_\nu^2(0.05)\)) | B1ft | |||||||||||||||||||||||||||||||||||
| \(2.75 \lt 11.070\), insufficient evidence to reject \(\mathrm{H}_0\) | M1 | |||||||||||||||||||||||||||||||||||
| Continuous uniform distribution is a suitable model | A1 | |||||||||||||||||||||||||||||||||||
| (12 marks) |
Notes
1st M1 for calculation of at least 3 widths and attempting proportions/probs. or for 1:2:3 ratio seen
1st A1 for correct probabilities
2nd A1 for all correct expected frequencies
2nd M1 for attempting \(\dfrac{(O - E)^2}{E}\) or \(\dfrac{O^2}{E}\), at least 3 correct expressions or values.
Follow through their \(E_i\) provided they are not all = 38
3rd A1 for a correct set of calcs - 3rd or 4th column. (2 dp or better and allow e.g. 0.94…)
3rd dM1 dependent on 2nd M1 for attempting a correct sum or calculation (must see at least 3 terms and +)
The first three Ms and As can be implied by a test statistic of awrt 2.75
4th M1 for a correct statement based on their test statistic ( > 1) and their cv (> 3.8)
Contradictory statements score M0 e.g. “significant” do not reject \(\mathrm{H}_0\).
5th A1 for a correct comment suggesting that continuous uniform model is suitable. No ft