| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | January |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Bivariate data |
| Type | Calculate regression line equation |
| Difficulty | Moderate -0.8 This is a standard S1 regression/correlation question requiring routine calculations using given summary statistics. Part (a) is trivial arithmetic, parts (b)-(c) apply standard formulas (Stt, Smm, Stm, then r), part (d) tests understanding that correlation is unaffected by linear transformations, and parts (e)-(f) require basic interpretation. All techniques are textbook exercises with no problem-solving or novel insight required, making it easier than average. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08b Linear coding: effect on pmcc5.08c Pearson: measure of straight-line fit |
| \(t\) (minutes) | £m |
| 15 | -3 |
| 23 | 17 |
| 5 | -19 |
| 16 | 4 |
| 30 | 12 |
| 6 | -9 |
| 32 | 27 |
| 23 | 6 |
| 35 | 20 |
| 27 | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Just 17 | B1 (1) | |
| (b) \(\sum t = 212\) and \(\sum m = 61\) (Accept as totals under each column in qu.) | B1, B1 | |
| \(S_m = 2485 - \frac{61 \times 212}{10} = 1191.8\) | M1, A1 | awrt 1190 or 119 (3sf) |
| \(S_n = 983.6\) (awrt 984) and \(S_{mm} = 1728.9\) (awrt 1730) | A1, A1 | (or 98.4 and 173) |
| Answer | Marks | Guidance |
|---|---|---|
| (c) \(r = \frac{1191.8}{\sqrt{983.6 \times 1728.9}} = 0.913922...\) | M1, A1 f.t. | awrt 0.914 |
| (d) 0.914 (Must be the same as (c) or awrt 0.914) | B1 f.t. ( | r |
| (e) 0.914 suggests longer spent shopping the more spent. (Idea more time, more spent) | B1 | |
| 0.178 different amounts spent for same time. | B1 | (2 marks) |
| (f) e.g. might spend short time buying 1 expensive item OR might spend a long time checking for bargains, talking, buying lots of cheap items. | B1g | (1 mark) |
**(a)** Just 17 | B1 (1) |
**(b)** $\sum t = 212$ and $\sum m = 61$ (Accept as totals under each column in qu.) | B1, B1 |
$S_m = 2485 - \frac{61 \times 212}{10} = 1191.8$ | M1, A1 | awrt 1190 or 119 (3sf)
$S_n = 983.6$ (awrt 984) and $S_{mm} = 1728.9$ (awrt 1730) | A1, A1 | (or 98.4 and 173)
(6 marks total)
**(c)** $r = \frac{1191.8}{\sqrt{983.6 \times 1728.9}} = 0.913922...$ | M1, A1 f.t. | awrt 0.914 | A1 | (3 marks)
**(d)** 0.914 (Must be the same as (c) or awrt 0.914) | B1 f.t. (|r| <1) | e.g. linear transformation, coding does not affect coefficient (or recalculate) | dB1 | (2 marks)
**(e)** 0.914 suggests longer spent shopping the more spent. (Idea more time, more spent) | B1 |
0.178 different amounts spent for same time. | B1 | (2 marks)
**(f)** e.g. might spend short time buying 1 expensive item OR might spend a long time checking for bargains, talking, buying lots of cheap items. | B1g | (1 mark)
**Total: 15 marks**
---\begin{enumerate}
\item As part of a statistics project, Gill collected data relating to the length of time, to the nearest minute, spent by shoppers in a supermarket and the amount of money they spent. Her data for a random sample of 10 shoppers are summarised in the table below, where $t$ represents time and $\pounds m$ the amount spent over $\pounds 20$.
\end{enumerate}
\begin{center}
\begin{tabular}{|l|l|}
\hline
$t$ (minutes) & £m \\
\hline
15 & -3 \\
\hline
23 & 17 \\
\hline
5 & -19 \\
\hline
16 & 4 \\
\hline
30 & 12 \\
\hline
6 & -9 \\
\hline
32 & 27 \\
\hline
23 & 6 \\
\hline
35 & 20 \\
\hline
27 & 6 \\
\hline
\end{tabular}
\end{center}
(a) Write down the actual amount spent by the shopper who was in the supermarket for 15 minutes.\\
(b) Calculate $S _ { t t } , S _ { m m }$ and $S _ { t m }$.
$$\text { (You may use } \Sigma t ^ { 2 } = 5478 \Sigma m ^ { 2 } = 2101 \Sigma t m = 2485 \text { ) }$$
(c) Calculate the value of the product moment correlation coefficient between $t$ and $m$.\\
(d) Write down the value of the product moment correlation coefficient between $t$ and the actual amount spent. Give a reason to justify your value.
On another day Gill collected similar data. For these data the product moment correlation coefficient was 0.178\\
(e) Give an interpretation to both of these coefficients.\\
(f) Suggest a practical reason why these two values are so different.
\hfill \mbox{\textit{Edexcel S1 2007 Q1 [15]}}