S3 June 2013 Q1
1. A doctor takes a random sample of 100 patients and measures their intake of saturated fats in their food and the level of cholesterol in their blood. The results are summarised in the table below.
| Cholesterol level Intake of saturated fats | High | Low |
|---|---|---|
| High | 12 | 8 |
| Low | 26 | 54 |
Using a 5% level of significance, test whether or not there is an association between cholesterol level and intake of saturated fats. State your hypotheses and show your working clearly. (10)
| Scheme | Marks | ||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| M1A1 (2) | ||||||||||||||||||||
| \(\mathrm{H}_0\) : Cholesterol level is independent of intake of saturated fats(no association) \(\mathrm{H}_1\) : Cholesterol level is not independent of intake of saturated fats (association) | B1 (1) | ||||||||||||||||||||
| dM1 A1 | ||||||||||||||||||||
| \(\displaystyle\sum\frac{(O - E)^2}{E} = 5.1358234..\) or \(\dfrac{12^2}{7.6} + \dfrac{8^2}{12.4} + \dfrac{26^2}{30.4} + \dfrac{54^2}{49.6} - 100 = 5.14\) (awrt 5.14) | A1 (3) | ||||||||||||||||||||
| \(\nu = (2 - 1)(2 - 1) = 1\) | B1 | ||||||||||||||||||||
| \(\chi^2_1(0.05) = 3.841\) | B1 (2) | ||||||||||||||||||||
| \(5.14 \gt 3.841\) so sufficient evidence to reject \(\mathrm{H}_0\) [Condone “accept \(\mathrm{H}_1\)”] | M1 | ||||||||||||||||||||
| Association between cholesterol level and saturated fat intake | A1 (2) | ||||||||||||||||||||
| (10 marks) |
Notes
Minimum working use part marks: \(E_i\) (2), Hyp (1), 5.14 (3), 3.841 (2), Conclusion (2)
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. Allow M1A0 for \(E_i\) rounded to integers
1st B1 for both hypotheses. Must mention “cholesterol” and “fats” at least once
Use of “relationship” or “correlation” or “connection” is B0
2nd dM1 for at least 2 correct terms (as in 3rd or 4th column) or correct expressions with their \(E_i\)
Dependent on 1st M1 Accept 2sf accuracy for the M mark
2nd A1 for all correct terms. May be implied by a correct ans.(2 dp or better)
Allow truncation eg 2.54… 3rd A1 for awrt 5.14
2nd B1 for correct degrees of freedom (may be implied by a cv of 3.841)
3rd M1 for a correct statement linking their test statistic and their cv(cv could be 2.705 or > 3.5)
Contradictory statements score M0 e.g. “significant, do not reject \(\mathrm{H}_0\)”
4th A1 for a correct comment in context - must mention “cholestorol” and “fats” condone “relationship” or “connection” here but not “correlation”.
e.g. “There is evidence of a relationship between cholesterol level and fat intake”
No follow through. If e.g hypotheses are the wrong way round A0 here.
(corrected from the printed mark scheme: the first term of the alternative calculation is printed as \(\frac{1.2^2}{7.6}\); the observed frequency is 12, so it is \(\frac{12^2}{7.6}\))