S3 June 2007 Q2
2. The Director of Studies at a large college believed that students’ grades in Mathematics were independent of their grades in English. She examined the results of a random group of candidates who had studied both subjects and she recorded the number of candidates in each of the 6 categories shown.
| Maths grade A or B | Maths grade C or D | Maths grade E or U | |
|---|---|---|---|
| English grade A or B | 25 | 25 | 10 |
| English grade C to U | 15 | 30 | 15 |
The Head of English suggested that the Director was losing accuracy by combining the English grades C to U in one row. He suggested that the Director should split the English grades into two rows, grades C or D and grades E or U as for Mathematics.
| Scheme | Marks | ||||||
|---|---|---|---|---|---|---|---|
| \(\mathrm{H}_0\) : Maths grades are independent of English grades or No association … \(\mathrm{H}_1\) : Maths and English grades are dependent or There is an association … | B1 | ||||||
Expected Frequencies e.g. \(\dfrac{60 \times 40}{120} = 20\)
| M1 A1 | ||||||
| \(\displaystyle\sum \frac{(O - E)^2}{E} = 2 \times \left(\frac{5^2}{20} + \frac{2.5^2}{27.5} + \frac{2.5^2}{12.5}\right),\ = 3.9545\ldots\) AWRT 3.95 or 3.955 | M1, A1 | ||||||
| \(\nu = (3 - 1)(2 - 1) = 2\); \(\chi^2_2(10\%)\) c.v. \(= 4.605\) | B1; B1 | ||||||
| \(3.95 \lt 4.605\) or not significant or do not reject \(\mathrm{H}_0\) (allow reject \(\mathrm{H}_1\)) | M1 | ||||||
| Insufficient evidence of an association between English and maths grades or there is support for the Director’s belief or Student’s grades in maths and English are independent | A1 | ||||||
| (9) |
Notes
1st B1 for both hypotheses in terms of independence or association and in context.
Must mention Maths and English in at least one of the hypotheses.
“relationship” or “correlation” or “connection” or “link” is B0
1st M1 for some correct calculation seen
1st A1 for all expected frequencies correct. Accept answers without formula seen.
2nd M1 for some evidence seen of attempt to calculate test statistic.
At least one correct term seen. Follow through their expected frequencies.
2nd A1 for AWRT 3.95. Answers only please escalate!
3rd M1 for correct comparison or statement – may be implied by correct conclusion.
3rd A1 for conclusion in context using “association” or “independence” in connection with grades.
Don’t insist on seeing English or maths mentioned here.
Use ISW for comments if a false statement and correct statement are seen.
| Scheme | Marks |
|---|---|
| May have some expected frequencies <5 (and hence need to pool rows/cols) | B1 |
| (1) | |
| (10 marks) |
Notes
B1 If they just say expected frequencies are “small” they must go onto mention need to pool.