| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Major (Further Statistics Major) |
| Year | 2022 |
| Session | June |
| Marks | 11 |
| 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 linear regression question from Further Maths Statistics requiring calculation of regression coefficients from raw data, computing residuals, and interpolation/extrapolation. While it involves more computation than typical A-level questions and requires understanding of residuals and reliability, the methods are entirely routine with no conceptual challenges or novel problem-solving required. The calculations are straightforward applications of standard formulas taught in the specification. |
| Spec | 5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line5.09e Use regression: for estimation in context |
| Temperature \(\left( t ^ { \circ } \mathrm { C } \right)\) | 20 | 22 | 24 | 26 | 28 | 30 | 32 | 34 | 36 |
| Tyre pressure \(( P\) bar \()\) | 2.012 | 2.036 | 2.065 | 2.074 | 2.114 | 2.140 | 2.149 | 2.176 | 2.192 |
| Temperature | 20 | 22 | 24 | 26 | 28 | 30 | 32 | 34 | 36 | ||
| - 0.003 | - 0.002 | 0.004 | - 0.010 | 0.011 | - 0.003 | 0.001 |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (a) | P = 0.01145t + 1.786 |
| Answer | Marks |
|---|---|
| [2] | 3.3 |
| 1.1 | For either 0.01145 or 1.786 (1.7858444) |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (b) | Residual for 28 = 2.114 – (0.01145 × 28 + 1.786) |
| Answer | Marks |
|---|---|
| Residual for 36 = –0.006 | M1 |
| Answer | Marks |
|---|---|
| [3] | 1.1a |
| Answer | Marks |
|---|---|
| 1.1 | Allow if wrong way around |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (c) | The fit is good as the residuals are all fairly small |
| Answer | Marks |
|---|---|
| is non-linear | E1 |
| Answer | Marks |
|---|---|
| [2] | 2.2a |
| 1.1 | Must be with reference to residuals |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (d) | Temperature 25°C Pressure ≈ 2.07(2) |
| Temperature 10°C Pressure ≈ 1.90(0) | B1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | FT awrt 2.07 |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (e) | Because the residuals are small, (and it is interpolation), |
| Answer | Marks |
|---|---|
| extrapolation. | E1 |
| Answer | Marks |
|---|---|
| [2] | 2.2a |
| 3.5b | Allow ‘As points lie close to the line’ |
Question 5:
5 | (a) | P = 0.01145t + 1.786 | M1
A1
[2] | 3.3
1.1 | For either 0.01145 or 1.786 (1.7858444)
BC Allow 3sf
5 | (b) | Residual for 28 = 2.114 – (0.01145 × 28 + 1.786)
or Residual for 36 = 2.192 – (0.01145 × 36 + 1.786)
Residual for 28 = 0.007
Residual for 36 = –0.006 | M1
A1
A1
[3] | 1.1a
1.1
1.1 | Allow if wrong way around
Allow 0.006 to 0.008 Allow without referring to part (a)
Allow −0.007 to −0.005
5 | (c) | The fit is good as the residuals are all fairly small
and there is no discernible pattern to suggest that the fit
is non-linear | E1
E1
[2] | 2.2a
1.1 | Must be with reference to residuals
Do not allow ‘Sum of residuals is small’
Allow ‘residuals not in blocks’
5 | (d) | Temperature 25°C Pressure ≈ 2.07(2)
Temperature 10°C Pressure ≈ 1.90(0) | B1
B1
[2] | 1.1
1.1 | FT awrt 2.07
FT awrt 1.90
If both given to 1dp allow MAX B0B1. If either given to more
than 3dp allow MAX B0B1
5 | (e) | Because the residuals are small, (and it is interpolation),
the first prediction likely to be reliable.
The second prediction is rather less reliable because it is
extrapolation. | E1
E1
[2] | 2.2a
3.5b | Allow ‘As points lie close to the line’
If only mention interpolation/extrapolation MAX E1E05 A motorist is investigating the relationship between tyre pressure and temperature. As the temperature increases during a hot day, she records the pressure (measured in bars) of one of her car tyres at specific temperatures of $20 ^ { \circ } \mathrm { C } , 22 ^ { \circ } \mathrm { C } , \ldots , 36 ^ { \circ } \mathrm { C }$. The results are shown in Table 5.1.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | }
\hline
Temperature $\left( t ^ { \circ } \mathrm { C } \right)$ & 20 & 22 & 24 & 26 & 28 & 30 & 32 & 34 & 36 \\
\hline
Tyre pressure $( P$ bar $)$ & 2.012 & 2.036 & 2.065 & 2.074 & 2.114 & 2.140 & 2.149 & 2.176 & 2.192 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 5.1}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Calculate the equation of the regression line of pressure on temperature. Give your answer in the form $P = a t + b$, giving the values of $a$ and $b$ to $\mathbf { 4 }$ significant figures.
\item Table 5.2 shows the residuals for most of the data values. Complete the copy of the table in the Printed Answer Booklet.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | }
\hline
Temperature & 20 & 22 & 24 & 26 & 28 & 30 & 32 & 34 & 36 \\
\hline
\begin{tabular}{ l }
Residual tyre \\
pressure \\
\end{tabular} & - 0.003 & - 0.002 & 0.004 & - 0.010 & & 0.011 & - 0.003 & 0.001 & \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 5.2}
\end{center}
\end{table}
\item With reference to the values of the residuals, comment on the goodness of fit of the regression line.
\item Use your answer to part (a) to calculate an estimate of the pressure in the tyre at each of the following temperatures, giving your answers to $\mathbf { 3 }$ decimal places.
\begin{itemize}
\item $25 ^ { \circ } \mathrm { C }$
\item $10 ^ { \circ } \mathrm { C }$
\item Comment on the reliability of each of your estimates.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Major 2022 Q5 [11]}}