| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2004 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Calculate y on x from raw data table |
| Difficulty | Moderate -0.8 This is a standard S1 linear regression question requiring routine application of formulae for Sxy, Sxx, correlation coefficient, and regression line. All necessary summations are provided, making it purely computational with no problem-solving insight required. The multi-part structure is typical but each step follows directly from textbook procedures. |
| Spec | 2.02c Scatter diagrams and regression lines5.08a Pearson correlation: calculate pmcc5.09b Least squares regression: concepts5.09c Calculate regression line5.09e Use regression: for estimation in context |
| \(h\) | 179 | 169 | 187 | 166 | 162 | 193 | 161 | 177 | 168 |
| \(c\) | 569 | 561 | 579 | 561 | 540 | 598 | 542 | 565 | 573 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Scatter diagram | B1 | Labels (not \(x,y\)) |
| B1 | Sensible scales; allow axis interchange | |
| B2 \((-1\) ee\()\) | Points (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(S_{hc}=884484-\frac{1562\times 5088}{9}=1433\frac{1}{3}\) | M1 | Correct use of \(S\) |
| A1 | \(1433\frac{1}{3}\); \(1433.\dot{3}\) | |
| \(S_{hh}=1000\frac{2}{9}\); \(S_{cc}=2550\) | A1; A1 | \(1000\frac{2}{9}\), \(1000.\dot{2}\); \(2550\) (4 marks) |
| (NB: accept \(-9\); i.e.: \(-159\frac{7}{27}\); \(111\frac{1}{81}\); \(283\frac{1}{3}\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(r=\frac{1433\frac{1}{3}}{\sqrt{1000\frac{2}{9}\times 2550}}\) | M1 | Substitution in correct formula |
| A1 ft | ||
| \(=0.897488\ldots\) | A1 | AWRT \(0.897\) (accept \(0.8975\)) (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Taller people tend to be more confident | B1 | Context (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(b=\frac{1433.\dot{3}}{1000.\dot{2}}=1.433014\ldots\) | M1 | |
| \(a=\frac{5088}{9}-\frac{1433.\dot{3}}{1000.\dot{2}}\times\frac{1562}{9}=316.6256\ldots\) | M1 | Allow use of their \(b\) |
| \(\therefore c=317+1.43h\) | A1 | 3sf (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(h=180\Rightarrow c=574.4\) or \(574.5683\ldots\) | M1 | Substitution of \(180\) |
| A1 | \(574\)–\(575\) (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(161\leq h\leq 193\) | B1 | (1 mark) |
## Question 2:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Scatter diagram | B1 | Labels (not $x,y$) |
| | B1 | Sensible scales; allow axis interchange |
| | B2 $(-1$ ee$)$ | Points **(4 marks)** |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $S_{hc}=884484-\frac{1562\times 5088}{9}=1433\frac{1}{3}$ | M1 | Correct use of $S$ |
| | A1 | $1433\frac{1}{3}$; $1433.\dot{3}$ |
| $S_{hh}=1000\frac{2}{9}$; $S_{cc}=2550$ | A1; A1 | $1000\frac{2}{9}$, $1000.\dot{2}$; $2550$ **(4 marks)** |
| (NB: accept $-9$; i.e.: $-159\frac{7}{27}$; $111\frac{1}{81}$; $283\frac{1}{3}$) | | |
### Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $r=\frac{1433\frac{1}{3}}{\sqrt{1000\frac{2}{9}\times 2550}}$ | M1 | Substitution in correct formula |
| | A1 ft | |
| $=0.897488\ldots$ | A1 | AWRT $0.897$ (accept $0.8975$) **(3 marks)** |
### Part (d)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Taller people tend to be more confident | B1 | Context **(1 mark)** |
### Part (e)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $b=\frac{1433.\dot{3}}{1000.\dot{2}}=1.433014\ldots$ | M1 | |
| $a=\frac{5088}{9}-\frac{1433.\dot{3}}{1000.\dot{2}}\times\frac{1562}{9}=316.6256\ldots$ | M1 | Allow use of their $b$ |
| $\therefore c=317+1.43h$ | A1 | 3sf **(3 marks)** |
### Part (f)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $h=180\Rightarrow c=574.4$ or $574.5683\ldots$ | M1 | Substitution of $180$ |
| | A1 | $574$–$575$ **(2 marks)** |
### Part (g)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $161\leq h\leq 193$ | B1 | **(1 mark)** |
**NB (a) No graph paper $\Rightarrow$ 0/4**
**(18 marks total)**
---2. A researcher thinks there is a link between a person's height and level of confidence. She measured the height $h$, to the nearest cm , of a random sample of 9 people. She also devised a test to measure the level of confidence $c$ of each person. The data are shown in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | }
\hline
$h$ & 179 & 169 & 187 & 166 & 162 & 193 & 161 & 177 & 168 \\
\hline
$c$ & 569 & 561 & 579 & 561 & 540 & 598 & 542 & 565 & 573 \\
\hline
\end{tabular}
\end{center}
[You may use $\Sigma h ^ { 2 } = 272094 , \Sigma c ^ { 2 } = 2878966 , \Sigma h c = 884484$ ]
\begin{enumerate}[label=(\alph*)]
\item Draw a scatter diagram to illustrate these data.
\item Find exact values of $S _ { h c } S _ { h h }$ and $S _ { c c }$.
\item Calculate the value of the product moment correlation coefficient for these data.
\item Give an interpretation of your correlation coefficient.
\item Calculate the equation of the regression line of $c$ on $h$ in the form $c = a + b h$.
\item Estimate the level of confidence of a person of height 180 cm .
\item State the range of values of $h$ for which estimates of $c$ are reliable.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2004 Q2 [18]}}