| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Bivariate data |
| Type | Calculate r from summary statistics |
| Difficulty | Moderate -0.3 This is a standard S1 correlation and regression question requiring routine application of formulas with given summary statistics. Part (i) involves calculating PMCC using the standard formula (straightforward but computational), part (ii) tests understanding of correlation vs causation (basic conceptual point), and part (iii) requires finding regression line of x on y and making a prediction. All techniques are textbook exercises with no novel insight required, though the calculation in part (i) is somewhat involved, making it slightly easier than average overall. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.09a Dependent/independent variables5.09c Calculate regression line5.09e Use regression: for estimation in context |
| Date | April 2008 | July 2008 | October 2008 | January 2009 | April 2009 | July 2009 | October 2009 | January 2010 | April 2010 | July 2010 |
| UR, \(x\)\% | 5.2 | 5.7 | 6.1 | 6.8 | 7.5 | 7.8 | 7.8 | 7.9 | 7.8 | 7.7 |
| CPI, \(y\)\% | 3.0 | 4.4 | 4.5 | 3.0 | 2.3 | 1.8 | 1.5 | 3.5 | 3.7 | 3.1 |
### (i)
**Answer:**
- $S_{xx} = 503.45 - \frac{70.3^2}{10} (= 9.241)$
- $S_{xy} = 103.94 - \frac{30.8^2}{10} (= 9.076)$
- $S_{yy} = 211.9 - \frac{70.3\times 30.8}{10} (= -4.624)$
- $r = \frac{"-4.624"}{\sqrt{9.241"\times"9.076"}} = -0.5049\ldots$ or $-0.505$ (3 sfs)
**Marks:** M1, M1, M1, A1[3]
**Guidance:**
- Correct sub in any correct $S$ formula
- Correct sub in any correct $r$ formula
- Must be correct sub in all $S$'s & $r$ but not nec'y accurate
### (ii)
**Answer:** Correlation (of UR & CPI) does not imply causation oe or r not close to –1
**Marks:** B1[1]
**Guidance:**
- Allow One may depend on another factor
- Allow without context
- NOT eg UR is independent
- NOT eg Only for the given years
- NOT eg Only for certain months
### (iii) (a)
**Answer:**
- $b = \frac{S_{xy}}{S_{yy}} = \frac{"-4.624"}{"-9.076"} (= \frac{-1156}{-2269}$ or $-0.50948)$
- $x - \frac{70.3}{10} = \frac{"-1156"}{-2269}(y - \frac{30.8}{10})$
- $x = -0.51y + 8.6$ (2 sfs) or $x = \frac{-1156}{2269}y + 8.6$
**Marks:** M1, M1, A1[3]
**Guidance:**
- If y on x: $b = \frac{S_{xx}}{S_{yy}} = \frac{"-4.624"}{"-9.241"} (= -0.500)$ M1
- If y on x found in (a): $b = -0.500x + 6.6$ M1, or $a' = \frac{"-1156"}{-2269}\times(-\frac{30.8}{10}) + \frac{70.3}{10}$ M1
- NB use $b' (= -0.509)$, not $r(= -0.5049)$ y = -0.50x + 6.6 A0
- y = -0.50x + 6.6 A0
### (iii) (b)
**Answer:** $x = -0.509 \times 4.0 + 8.60 = 6.56$ (3 sf) or $6.6$ (2 sf)
**Marks:** M1, A1ft[2]
**Guidance:**
- Allow sub $y = 0.04$ for M1 only
- ft their eqn; ans to 2 sf A1ft
---The table shows some of the values of the seasonally adjusted Unemployment Rate (UR), $x$\%, and the Consumer Price Index (CPI), $y$\%, in the United Kingdom from April 2008 to July 2010.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Date & April 2008 & July 2008 & October 2008 & January 2009 & April 2009 & July 2009 & October 2009 & January 2010 & April 2010 & July 2010 \\
\hline
UR, $x$\% & 5.2 & 5.7 & 6.1 & 6.8 & 7.5 & 7.8 & 7.8 & 7.9 & 7.8 & 7.7 \\
\hline
CPI, $y$\% & 3.0 & 4.4 & 4.5 & 3.0 & 2.3 & 1.8 & 1.5 & 3.5 & 3.7 & 3.1 \\
\hline
\end{tabular}
These data are summarised below.
$$n = 10 \quad \sum x = 70.3 \quad \sum x^2 = 503.45 \quad \sum y = 30.8 \quad \sum y^2 = 103.94 \quad \sum xy = 211.9$$
\begin{enumerate}[label=(\roman*)]
\item Calculate the product moment correlation coefficient, $r$, for the data, showing that $-0.6 < r < -0.5$. [3]
\item Karen says "The negative value of $r$ shows that when the Unemployment Rate increases, it causes the Consumer Price Index to decrease." Give a criticism of this statement. [1]
\item \begin{enumerate}[label=(\alph*)]
\item Calculate the equation of the regression line of $x$ on $y$. [3]
\item Use your equation to estimate the value of the Unemployment Rate in a month when the Consumer Price Index is 4.0\%. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR S1 2013 Q5 [9]}}