S1 January 2011 Q1
1. A random sample of 50 salmon was caught by a scientist. He recorded the length \(l\) cm and weight \(w\) kg of each salmon.
The following summary statistics were calculated from these data.
\[\sum l = 4027 \qquad \sum l^2 = 327\,754.5 \qquad \sum w = 357.1 \qquad \sum lw = 29\,330.5 \qquad S_{ww} = 289.6\]| Scheme | Marks |
|---|---|
| \(S_{ll} = 327754.5 - \dfrac{4027^2}{50} = 3419.92\) | M1 A1 |
| \(S_{lw} = 29330.5 - \dfrac{357.1 \times 4027}{50} = 569.666\) | A1 |
| (3) |
Notes
M1 for at least one correct expression
1st A1 for \(S_{ll} =\) awrt 3420 (Condone \(S_{xx} = \ldots\) or even \(S_{yy} = \ldots\))
2nd A1 for \(S_{lw} =\) awrt 570 (Condone \(S_{xy} = \ldots\) )
| Scheme | Marks |
|---|---|
| \(r = \dfrac{569.666}{\sqrt{3419.92 \times 289.6}} = 0.572\) awrt 0.572 or 0.573 | M1 A1 |
| (2) |
Notes
M1 for attempt at correct formula.
Must have their \(S_{ll}\), \(S_{lw}\) and given \(S_{ww}\) in the correct places
If \(S_{ll}\), \(S_{lw}\) are correct and an answer of awrt 0.57 is seen then award M1A0
M0 for \(\dfrac{29330.5}{\sqrt{327754.5 \times 289.6}}\)
| Scheme | Marks |
|---|---|
| As the length of the salmon increases the weight increases | B1ft |
| (1) | |
| (6 marks) |
Notes
B1ft for a comment mentioning “length” and “weight”, not just \(l\) and \(w\), and the idea of longer salmon weighing more.
e.g. “positive correlation between weight and length” is B0 since the idea of positive correlation is not explained.
Allow “larger” instead of “heavier” or “longer”
Ignore any spurious values mentioned such as 0.572
If their \(r\) is negative (but must be \(r \gt -1\)) ft an appropriate comment.
Condone \(r \gt 1\) if comment is correct.
If \(|r| \lt 0.4\) allow a comment of no or little relationship between weight and length but for \(0 \lt r \lt 0.4\) the printed answer is still acceptable too.
Treat mention of “skewness” as ISW if a correct interpretation is given