S3 June 2010 Q5
5. A random sample of 100 people were asked if their finances were worse, the same or better than this time last year. The sample was split according to their annual income and the results are shown in the table below.
| Finances Annual income | Worse | Same | Better |
|---|---|---|---|
| Under £15 000 | 14 | 11 | 9 |
| £15 000 and above | 17 | 20 | 29 |
Test, at the 5% level of significance, whether or not the relative state of their finances is independent of their income range. State your hypotheses and show your working clearly. (10)
| Scheme | Marks | ||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| M1 A1 | ||||||||||||||||||||||||||||
| \(\mathrm{H}_0\) : State of finances and income are independent (not associated) \(\mathrm{H}_1\) : State of finances and income are not independent (associated) | B1 | ||||||||||||||||||||||||||||
| M1 A1 | ||||||||||||||||||||||||||||
| \(\displaystyle\sum\frac{(O_i - E_i)^2}{E_i} = 3.553\ldots\) or \(\displaystyle\sum\frac{O_i^2}{E_i} - 100 = 103.553\ldots - 100 = 3.553\ldots\) (awrt 3.55) | A1 | ||||||||||||||||||||||||||||
| \(\nu = (3 - 1)(2 - 1) = 2\) | B1 | ||||||||||||||||||||||||||||
| cv is 5.991 | B1 | ||||||||||||||||||||||||||||
| \(3.553 \lt 5.991\) so insufficient evidence to reject \(\mathrm{H}_0\) or not significant | M1 | ||||||||||||||||||||||||||||
| There is no evidence of association between state of finances and income. | A1 | ||||||||||||||||||||||||||||
| (10 marks) |
Notes
1st M1 for some use of \(\dfrac{\text{Row Total} \times \text{Col.Total}}{\text{Grand Total}}\). May be implied by correct \(E_i\)
1st A1 for all expected frequencies correct
B1 for both hypotheses. Must mention “state” or “finances” and “income” at least once
Use of “relationship” or “correlation” or “connection” is B0
2nd M1 for at least two correct terms (as in 3rd or 4th column) or correct expressions with their \(E_i\)
2nd A1 for all correct terms. May be implied by a correct answer.(2 dp or better-allow eg 1.13…)
3rd M1 for a correct statement linking their test statistic and their cv. Must be \(\chi^2\) not normal.
4th A1 for a correct comment in context - must mention “state” or “finances” and “income” condone “relationship” or “connection” here but not “correlation”. No follow through.
e.g. “There is no evidence of a relationship between finances and income”