| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Find means from regression lines |
| Difficulty | Moderate -0.8 This is a straightforward regression question testing basic recall of key properties: (i) recognizing negative correlation from negative gradients, (ii) substituting into the appropriate regression line, and (iii) using the standard result that both regression lines pass through (x̄, ȳ). All parts are routine applications of textbook knowledge with minimal calculation required. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.09a Dependent/independent variables5.09c Calculate regression line |
| Answer | Marks | Guidance |
|---|---|---|
| 1(i) | Negative, because (grad or coeff of \(x\) in 1st eqn or \(x\)-value or reg coeff of \(B\) or \(-0.6\)) is negative | B1: 1 |
| 1(ii) | \(x = -1.6 \times 7.0 + 21\); \(x = 9.8\) | M1: 1; A1: 2 |
| 1(iii) | \(y = -0.6(-1.6y + 21) + 13\) or similar; \(\bar{x} = 5, \bar{y} = 10\) | M1: 1; A1A1: 3 |
1(i) | Negative, because (grad or coeff of $x$ in 1st eqn or $x$-value or reg coeff of $B$ or $-0.6$) is negative | B1: 1 | Neg because $x$ incr & $y$ decr |
1(ii) | $x = -1.6 \times 7.0 + 21$; $x = 9.8$ | M1: 1; A1: 2 | Sub $y=7.0$ in 2nd eqn. Allow 1 sign error. If sub in both must choose 2nd |
1(iii) | $y = -0.6(-1.6y + 21) + 13$ or similar; $\bar{x} = 5, \bar{y} = 10$ | M1: 1; A1A1: 3 | Obtain correct eqn in 1 variable. Allow 1 num'l error. Allow without bars |
**Total: 6**
---1 Some observations of bivariate data were made and the equations of the two regression lines were found to be as follows.
$$\begin{array} { c c }
y \text { on } x : & y = - 0.6 x + 13.0 \\
x \text { on } y : & x = - 1.6 y + 21.0
\end{array}$$
(i) State, with a reason, whether the correlation between $x$ and $y$ is negative or positive.\\
(ii) Neither variable is controlled. Calculate an estimate of the value of $x$ when $y = 7.0$.\\
(iii) Find the values of $\bar { x }$ and $\bar { y }$.
\hfill \mbox{\textit{OCR S1 2006 Q1 [6]}}