| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Topic | Bivariate data |
| Type | Hypothesis test for correlation |
| Difficulty | Moderate -0.3 This is a straightforward statistics question requiring standard techniques: finding median/IQR, applying the 1.5×IQR outlier rule, calculating PMCC using a formula or calculator, and making basic comments about correlation vs causation. All parts are routine A-level statistics procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.02f Measures of average and spread2.02h Recognize outliers5.08a Pearson correlation: calculate pmcc |
| 15 | 16 | 18 | 19 | 21 | 36 | 36 | 38 | 41 | 41 |
| 43 | 47 | 51 | 55 | 56 | 60 | 62 | 64 | 110 |
| Salary in £10 000's | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| Self-esteem | 4 | 3 | 5 | 1 | 7 | 7 | 8 | 5 | 10 | 7 | 9 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) (i) Median = 41 or £41 000 | B1 | |
| Attempt to find UQ and LQ | M1 | |
| 56 − 21 = 35 (000s) | A1 | [3] |
| (ii) Find (their) UQ + 1.5(their IQR) | M1 | |
| State 108.5 < 110 so an outlier but retain as no evidence it is not genuine data | A1 | [2] |
| (b) (i) Attempt \(\frac{\sum xy - \frac{\sum x \sum y}{n}}{\sqrt{\left(\sum x^2 - \frac{(\sum x)^2}{n}\right)\left(\sum y^2 - \frac{(\sum y)^2}{n}\right)}}\) aef maybe implied | M1 | |
| 0.73 or rounding to 0.73 | A1 | |
| Positive correlation between salary and self-esteem | Dep A1 | [3] |
| (ii) High salaries do not cause raised self esteem | B1 | |
| Extrapolation to higher salaries than presented by the data invalid | B1 | [2] |
(a) (i) Median = 41 or £41 000 | B1 |
Attempt to find UQ and LQ | M1 |
56 − 21 = 35 (000s) | A1 | [3]
(ii) Find (their) UQ + 1.5(their IQR) | M1 |
State 108.5 < 110 so an outlier but retain as no evidence it is not genuine data | A1 | [2]
(b) (i) Attempt $\frac{\sum xy - \frac{\sum x \sum y}{n}}{\sqrt{\left(\sum x^2 - \frac{(\sum x)^2}{n}\right)\left(\sum y^2 - \frac{(\sum y)^2}{n}\right)}}$ aef maybe implied | M1 |
0.73 or rounding to 0.73 | A1 |
Positive correlation between salary and self-esteem | Dep A1 | [3]
(ii) High salaries do not cause raised self esteem | B1 |
Extrapolation to higher salaries than presented by the data invalid | B1 | [2]A survey was conducted into the annual salary offered for 19 different jobs in 2008. The results were as follows, in thousands of pounds.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
15 & 16 & 18 & 19 & 21 & 36 & 36 & 38 & 41 & 41 \\
43 & 47 & 51 & 55 & 56 & 60 & 62 & 64 & 110 & \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\begin{enumerate}[label=(\roman*)]
\item Find the median and interquartile range of this set of data. [3]
\item Show that £110 000 is an outlier and discuss briefly how this outlier might be treated. [2]
\end{enumerate}
\end{enumerate}
It was decided to undertake a further study to see if self-esteem was correlated with level of annual salary. A random sample of 11 employees was taken and self-esteem was rated on a scale of 1 to 10 with the highest self-esteem being 10.
The results were as follows.
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Salary in £10 000's & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\
\hline
Self-esteem & 4 & 3 & 5 & 1 & 7 & 7 & 8 & 5 & 10 & 7 & 9 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\begin{enumerate}[label=(\roman*)]
\item Calculate the product-moment correlation coefficient and comment on the result. [3]
\item "The best way to increase self-esteem is to pay people more money." Comment on this claim. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2010 Q13 [10]}}