S1 January 2013 Q1
1. A teacher asked a random sample of 10 students to record the number of hours of television, \(t\), they watched in the week before their mock exam. She then calculated their grade, \(g\), in their mock exam. The results are summarised as follows.
\[\sum t = 258 \qquad \sum t^2 = 8702 \qquad \sum g = 63.6 \qquad \mathrm{S}_{gg} = 7.864 \qquad \sum gt = 1550.2\]The teacher also recorded the number of hours of revision, \(v\), these 10 students completed during the week before their mock exam. The correlation coefficient between \(t\) and \(v\) was \(-0.753\)
| Scheme | Marks |
|---|---|
| \((\mathrm{S}_{tt}) = 8702 - \dfrac{258^2}{10}\) or \((\mathrm{S}_{gt}) = 1550.2 - \dfrac{258\times 63.6}{10}\) | M1 |
| \((\mathrm{S}_{tt} =)\,2045.6\), \((\mathrm{S}_{gt} =) -90.68\) awrt (2046), awrt \(-90.7\) | A1, A1 |
| (3) |
Notes
M1 for at least one correct expression
1st A1 for \(\mathrm{S}_{tt} =\) awrt 2046 (Condone \(S_{xx} = \ldots\) or even \(S_{yy} = \ldots\))
2nd A1 for \(\mathrm{S}_{gt} =\) awrt \(-90.7\) (Condone \(S_{xy} = \ldots\))
| Scheme | Marks |
|---|---|
| \(r = \dfrac{-90.68}{\sqrt{2045.6\times 7.864}} = -0.714956\ldots\) awrt \(-0.715\) | M1 A1 |
| (2) |
Notes
M1 for attempt at correct formula. Must have their \(\mathrm{S}_{tt}\), \(\mathrm{S}_{gt}\) and given \(\mathrm{S}_{gg}\) in the correct places. Condone missing “−”
Award M1A0 for awrt \(-0.71\) with no expression seen
M0 for \(\dfrac{1550.2}{\sqrt{8702\times 7.864}}\)
Correct answer only is 2/2
| Scheme | Marks |
|---|---|
| Positive | B1 |
| e.g. high \(v\) corresponds to low \(t\) and low \(t\) corresponds to high \(g\) so expect high \(v\) to corresponds to high \(g\) or expect more revision to result in a better grade | B1 |
| (2) | |
| (7 marks) |
Notes
1st B1 for saying "positive". Ignore mention of skew.
2nd B1 for suitable reason that mentions at least \(v\) and \(g\) and supports positive correlation.
e.g. “the less revision done the lower the grade” is B1
“should do better with more revision” is B0 since does not mention grades
“both coefficients are similar” or two sketches of negative correlation with labelled axes is B1 since \(v\), \(t\) and \(g\) are implied
Allow use of letters \(v\) and \(g\)
Allow equivalent terms e.g. “study” instead of “revision” or “score” instead of “grade”