| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2009 |
| Session | January |
| Marks | 8 |
| 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 straightforward application of standard linear regression formulas with all summary statistics provided. Students substitute into formulas for gradient and intercept, then use the equation for predictions. The only mild challenge is the commentary on extrapolation reliability, but with r=0.988 given, this requires minimal statistical insight. Significantly easier than average A-level questions. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.09a Dependent/independent variables5.09c Calculate regression line5.09e Use regression: for estimation in context |
| Age \(( x\) years \()\) | 11 | 12 | 20 | 28 | 35 | 45 | 51 |
| Mean trunk diameter \(( y \mathrm {~cm} )\) | 12.2 | 16.0 | 26.4 | 39.2 | 39.6 | 51.3 | 60.6 |
2 The table shows the age, $x$ years, and the mean diameter, $y \mathrm {~cm}$, of the trunk of each of seven randomly selected trees of a certain species.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | }
\hline
Age $( x$ years $)$ & 11 & 12 & 20 & 28 & 35 & 45 & 51 \\
\hline
Mean trunk diameter $( y \mathrm {~cm} )$ & 12.2 & 16.0 & 26.4 & 39.2 & 39.6 & 51.3 & 60.6 \\
\hline
\end{tabular}
\end{center}
$$\left[ n = 7 , \Sigma x = 202 , \Sigma y = 245.3 , \Sigma x ^ { 2 } = 7300 , \Sigma y ^ { 2 } = 10510.65 , \Sigma x y = 8736.9 . \right]$$
(i) (a) Use an appropriate formula to show that the gradient of the regression line of $y$ on $x$ is 1.13 , correct to 2 decimal places.\\
(b) Find the equation of the regression line of $y$ on $x$.\\
(ii) Use your equation to estimate the mean trunk diameter of a tree of this species with age\\
(a) 30 years,\\
(b) 100 years.
It is given that the value of the product moment correlation coefficient for the data in the table is 0.988 , correct to 3 decimal places.\\
(iii) Comment on the reliability of each of your two estimates.
\hfill \mbox{\textit{OCR S1 2009 Q2 [8]}}