| Exam Board | OCR MEI |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | January |
| Marks | 19 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Linearize non-linear relationships |
| Difficulty | Standard +0.3 This is a straightforward linear regression question requiring standard calculations (scatter diagram, regression line equation, predictions, residuals) with routine interpretation. While it involves multiple parts, each step follows textbook procedures with no conceptual challenges or novel problem-solving required. The linearization aspect mentioned in the topic is not actually tested here—the relationship appears linear as given. Slightly above average difficulty only due to the number of parts and need for careful arithmetic. |
| Spec | 5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line5.09e Use regression: for estimation in context |
| Thickness \(( t \mathrm {~mm} )\) | 20 | 40 | 60 | 80 | 100 |
| Maximum height \(( h \mathrm {~m} )\) | 0.72 | 1.09 | 1.62 | 1.97 | 2.34 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((0.0206 \times 70) + 0.312 = 1.754\) | B1 | Allow 1.75; FT their equation provided \(b > 0\) |
| Likely to be reliable as interpolation | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((0.0206 \times 120) + 0.312 = 2.784\) | B1 | Allow 2.78; FT their equation provided \(b > 0\) |
| Could be unreliable as extrapolation | E1 | Condone "reliable as 120 is not too far away from the data used to produce the equation" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Thickness \(= 40 \Rightarrow\) predicted max height \(= (0.0206 \times 40) + 0.312 = 1.136\) | M1 | For prediction; FT their equation provided \(b > 0\) |
| Residual \(= 1.09 - 1.136\) | M1 | For difference between 1.09 and prediction |
| \(= -0.046\) | A1 | Allow \(-0.05\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Regression line gives a prediction of \((0.0206 \times 200) + 0.312 = 4.432\) | B1* | B1 for obtaining a prediction from regression equation or from graph |
| This is well above the observed value | E1 dep* | E1 for noting the large difference between prediction and actual value |
| It could be that the relationship breaks down for larger thickness, or that the relationship is not linear | E1 | E1 for suitable interpretation regarding the relationship between maximum height and thickness |
# Question 1:
## Part (iv)(A)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(0.0206 \times 70) + 0.312 = 1.754$ | B1 | Allow 1.75; FT their equation provided $b > 0$ |
| Likely to be reliable as interpolation | E1 | |
## Part (iv)(B)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(0.0206 \times 120) + 0.312 = 2.784$ | B1 | Allow 2.78; FT their equation provided $b > 0$ |
| Could be unreliable as extrapolation | E1 | Condone "reliable as 120 is not too far away from the data used to produce the equation" |
## Part (v)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Thickness $= 40 \Rightarrow$ predicted max height $= (0.0206 \times 40) + 0.312 = 1.136$ | M1 | For prediction; FT their equation provided $b > 0$ |
| Residual $= 1.09 - 1.136$ | M1 | For difference between 1.09 and prediction |
| $= -0.046$ | A1 | Allow $-0.05$ |
## Part (vi)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Regression line gives a prediction of $(0.0206 \times 200) + 0.312 = 4.432$ | B1* | B1 for obtaining a prediction from regression equation or from graph |
| This is well above the observed value | E1 dep* | E1 for noting the large difference between prediction and actual value |
| It could be that the relationship breaks down for larger thickness, or that the relationship is not linear | E1 | E1 for suitable interpretation regarding the relationship between maximum height and thickness |
---1 A manufacturer of playground safety tiles is testing a new type of tile. Tiles of various thicknesses are tested to estimate the maximum height at which people would be unlikely to sustain injury if they fell onto a tile. The results of the test are as follows.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Thickness $( t \mathrm {~mm} )$ & 20 & 40 & 60 & 80 & 100 \\
\hline
Maximum height $( h \mathrm {~m} )$ & 0.72 & 1.09 & 1.62 & 1.97 & 2.34 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\roman*)]
\item Draw a scatter diagram to illustrate these data.
\item State which of the two variables is the independent variable, giving a reason for your answer.
\item Calculate the equation of the regression line of maximum height on thickness.
\item Use the equation of the regression line to calculate estimates of the maximum height for thicknesses of\\
(A) 70 mm ,\\
(B) 120 mm .
Comment on the reliability of each of these estimates.
\item Calculate the value of the residual for the data point at which $t = 40$.
\item In a further experiment, the manufacturer tests a tile with a thickness of 200 mm and finds that the corresponding maximum height is 2.96 m . What can be said about the relationship between tile thickness and maximum height?
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S2 2013 Q1 [19]}}