| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2001 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Calculate y on x from raw data table |
| Difficulty | Moderate -0.3 This is a standard S1 regression question requiring calculation of the regression line from summary statistics (which are provided), plotting a scatter diagram, and interpreting the model. The calculations are routine using standard formulas, and the interpretation/justification parts require only basic understanding of regression limitations. Slightly easier than average due to the provided summary statistics and straightforward context. |
| Spec | 2.02c Scatter diagrams and regression lines5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line5.09e Use regression: for estimation in context |
| \(x\) | 12 | 15 | 7 | 11 | 1 | 8 | 4 | 6 | 9 | 3 |
| \(y\) | 8 | 4 | 13 | 8 | 18 | 12 | 15 | 14 | 12 | 16 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Scatter diagram with scales and labels | B1 | Scales & labels |
| Points plotted correctly | B2 | \((9, 8 \text{ points } B1)\) (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sum x = 76\), \(\sum y = 120\) | B1, B1 (2) | Can be implied |
| \(b = \frac{10 \times 749 - 76 \times 120}{10 \times 746 - (76)^2} = \frac{-1630}{1684}\) | M1 | Use of \(S_{xy}/S_{xx}\) a.e.f. |
| Correct substitute | A1 | |
| \(= -0.96793\ldots\) | A1 (3) | AWRT \(-0.97\) |
| \(a = \frac{120}{10} - (-0.96793\ldots)\left(\frac{76}{10}\right)\) | M1 | Use of \(\bar{y} - b\bar{x}\) |
| \(= 19.356\ldots\) | A1, A1 (3) | Correct subst. without rounding approx.; AWRT \(19.4\) |
| \(\therefore y = 19.4 - 0.968x\) or \(19.4 - 0.97x\) | B1 (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(b \Rightarrow\) for every extra hour of practice, \(1\) (\(-0.968\)) less error will be made | B1\(\checkmark\) | |
| \(a \Rightarrow\) without practice, \(19/20\) errors will be made | B1\(\checkmark\) (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) Yes — all points reasonably close to the line | B1 | |
| (ii) No — more likely to be \(\leq\) | B1 (2) |
## Question 7:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Scatter diagram with scales and labels | B1 | Scales & labels |
| Points plotted correctly | B2 | $(9, 8 \text{ points } B1)$ (3) |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sum x = 76$, $\sum y = 120$ | B1, B1 (2) | Can be implied |
| $b = \frac{10 \times 749 - 76 \times 120}{10 \times 746 - (76)^2} = \frac{-1630}{1684}$ | M1 | Use of $S_{xy}/S_{xx}$ a.e.f. |
| Correct substitute | A1 | |
| $= -0.96793\ldots$ | A1 (3) | AWRT $-0.97$ |
| $a = \frac{120}{10} - (-0.96793\ldots)\left(\frac{76}{10}\right)$ | M1 | Use of $\bar{y} - b\bar{x}$ |
| $= 19.356\ldots$ | A1, A1 (3) | Correct subst. without rounding approx.; AWRT $19.4$ |
| $\therefore y = 19.4 - 0.968x$ or $19.4 - 0.97x$ | B1 (1) | |
### Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $b \Rightarrow$ for every extra hour of practice, $1$ ($-0.968$) less error will be made | B1$\checkmark$ | |
| $a \Rightarrow$ without practice, $19/20$ errors will be made | B1$\checkmark$ (2) | |
### Part (d)
| Answer/Working | Mark | Guidance |
|---|---|---|
| (i) Yes — all points reasonably close to the line | B1 | |
| (ii) No — more likely to be $\leq$ | B1 (2) | |7. A music teacher monitored the sight-reading ability of one of her pupils over a 10 week period. At the end of each week, the pupil was given a new piece to sight-read and the teacher noted the number of errors $y$. She also recorded the\\
number of hours $x$ that the pupil had practised each week. The data are shown in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | c | }
\hline
$x$ & 12 & 15 & 7 & 11 & 1 & 8 & 4 & 6 & 9 & 3 \\
\hline
$y$ & 8 & 4 & 13 & 8 & 18 & 12 & 15 & 14 & 12 & 16 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Plot these data on a scatter diagram.
\item Find the equation of the regression line of $y$ on $x$ in the form $y = a + b x$.
$$\text { (You may use } \left. \Sigma x ^ { 2 } = 746 , \Sigma x y = 749 . \right)$$
\item Give an interpretation of the slope and the intercept of your regression line.
\item State whether or not you think the regression model is reasonable
\begin{enumerate}[label=(\roman*)]
\item for the range of $x$-values given in the table,
\item for all possible $x$-values.
In each case justify your answer either by giving a reason for accepting the model or by suggesting an alternative model.
END
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2001 Q7 [16]}}