| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Bivariate data |
| Type | Calculate summary statistics (Sxx, Syy, Sxy) |
| Difficulty | Easy -1.2 This is a straightforward computational question requiring direct application of standard formulas for Sxx, Sxy, and correlation coefficient. All necessary summary statistics are provided, requiring only substitution into memorized formulas with no problem-solving or interpretation beyond a standard statement about correlation strength. |
| Spec | 5.08a Pearson correlation: calculate pmcc |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(S_{ll} = 327754.5 - \frac{4027^2}{50} = 3419.92\) | M1 A1 | M1 for at least one correct expression; 1st A1 for \(S_{ll}\) = awrt 3420 (Condone \(S_{xx}=\ldots\) or \(S_{yy}=\ldots\)) |
| \(S_{lw} = 29330.5 - \frac{357.1 \times 4027}{50} = 569.666\) | A1 | 2nd A1 for \(S_{lw}\) = awrt 570 (Condone \(S_{xy}=\ldots\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(r = \frac{569.666}{\sqrt{3419.92 \times 289.6}} = 0.572\) | M1 A1 | M1 for attempt at correct formula; must have their \(S_{ll}\), \(S_{lw}\) and given \(S_{ww}\) in correct places; awrt 0.572 or 0.573; M0 for \(\frac{29330.5}{\sqrt{327754.5 \times 289.6}}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| As the length of the salmon increases the weight increases | B1ft | Must mention "length" and "weight" (not just \(l\) and \(w\)) and idea of longer salmon weighing more; "positive correlation between weight and length" is B0; allow "larger" instead of "heavier" or "longer"; if \( |
# Question 1:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $S_{ll} = 327754.5 - \frac{4027^2}{50} = 3419.92$ | M1 A1 | M1 for at least one correct expression; 1st A1 for $S_{ll}$ = awrt 3420 (Condone $S_{xx}=\ldots$ or $S_{yy}=\ldots$) |
| $S_{lw} = 29330.5 - \frac{357.1 \times 4027}{50} = 569.666$ | A1 | 2nd A1 for $S_{lw}$ = awrt 570 (Condone $S_{xy}=\ldots$) |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $r = \frac{569.666}{\sqrt{3419.92 \times 289.6}} = 0.572$ | M1 A1 | M1 for attempt at correct formula; must have their $S_{ll}$, $S_{lw}$ and given $S_{ww}$ in correct places; awrt 0.572 or 0.573; M0 for $\frac{29330.5}{\sqrt{327754.5 \times 289.6}}$ |
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| As the length of the salmon increases the weight increases | B1ft | Must mention "length" and "weight" (not just $l$ and $w$) and idea of longer salmon weighing more; "positive correlation between weight and length" is B0; allow "larger" instead of "heavier" or "longer"; if $|r| < 0.4$ allow comment of no/little relationship |
---\begin{enumerate}
\item A random sample of 50 salmon was caught by a scientist. He recorded the length $l \mathrm {~cm}$ and weight $w \mathrm {~kg}$ of each salmon.
\end{enumerate}
The following summary statistics were calculated from these data.\\
$\sum l = 4027 \quad \sum l ^ { 2 } = 327754.5 \quad \sum w = 357.1 \quad \sum l w = 29330.5 \quad S _ { w w } = 289.6$\\
(a) Find $S _ { l l }$ and $S _ { l w }$\\
(b) Calculate, to 3 significant figures, the product moment correlation coefficient between $l$ and $w$.\\
(c) Give an interpretation of your coefficient.\\
\hfill \mbox{\textit{Edexcel S1 2011 Q1 [6]}}