| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Hypothesis test of Pearson’s product-moment correlation coefficient |
| Type | Describe correlation from scatter diagram |
| Difficulty | Moderate -0.8 This question tests basic interpretation of scatter diagrams and understanding of correlation hypothesis testing. Part (a) requires simple ratio estimation from a graph, part (b)(i) is straightforward table lookup, part (b)(ii) tests understanding that biased sampling affects validity (a standard critique), and parts (c)-(d) require simple contextual interpretation. While it's a multi-part question, each component involves routine application of statistical concepts without requiring novel insight or complex reasoning. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| \multirow{2}{*}{1-tail test 2-tail test} | 5\% | 2.5\% | 1\% | 0.5\% |
| 10\% | 5\% | 2\% | 1\% | |
| \(n\) | ||||
| 1 | - | - | - | - |
| 2 | - | - | - | - |
| 3 | 0.9877 | 0.9969 | 0.9995 | 0.9999 |
| 4 | 0.9000 | 0.9500 | 0.9800 | 0.9900 |
| 5 | 0.8054 | 0.8783 | 0.9343 | 0.9587 |
| 6 | 0.7293 | 0.8114 | 0.8822 | 0.9172 |
| 7 | 0.6694 | 0.7545 | 0.8329 | 0.8745 |
| 8 | 0.6215 | 0.7067 | 0.7887 | 0.8343 |
| 9 | 0.5822 | 0.6664 | 0.7498 | 0.7977 |
| 10 | 0.5494 | 0.6319 | 0.7155 | 0.7646 |
| 11 | 0.5214 | 0.6021 | 0.6851 | 0.7348 |
| 12 | 0.4973 | 0.5760 | 0.6581 | 0.7079 |
| 13 | 0.4762 | 0.5529 | 0.6339 | 0.6835 |
| 14 | 0.4575 | 0.5324 | 0.6120 | 0.6614 |
| 15 | 0.4409 | 0.5140 | 0.5923 | 0.6411 |
| 16 | 0.4259 | 0.4973 | 0.5742 | 0.6226 |
| 17 | 0.4124 | 0.4821 | 0.5577 | 0.6055 |
| 18 | 0.4000 | 0.4683 | 0.5425 | 0.5897 |
| 19 | 0.3887 | 0.4555 | 0.5285 | 0.5751 |
| 20 | 0.3783 | 0.4438 | 0.5155 | 0.5614 |
| 21 | 0.3687 | 0.4329 | 0.5034 | 0.5487 |
| 22 | 0.3598 | 0.4227 | 0.4921 | 0.5368 |
| 23 | 0.3515 | 0.4132 | 0.4815 | 0.5256 |
| 24 | 0.3438 | 0.4044 | 0.4716 | 0.5151 |
| 25 | 0.3365 | 0.3961 | 0.4622 | 0.5052 |
| 26 | 0.3297 | 0.3882 | 0.4534 | 0.4958 |
| 27 | 0.3233 | 0.3809 | 0.4451 | 0.4869 |
| 28 | 0.3172 | 0.3739 | 0.4372 | 0.4785 |
| 29 | 0.3115 | 0.3673 | 0.4297 | 0.4705 |
| 30 | 0.3061 | 0.3610 | 0.4226 | 0.4629 |
| Answer | Marks | Guidance |
|---|---|---|
| \(k > 1.4\) (allow \(k > 1.1\) to \(1.6\)); \(k < 0.25\) (allow \(k < 0.2\) to \(0.3\)) | B1, B1 [2] | Allow \(\geq\) and \(\leq\); SC: \(0.25 < k < 1.4\): B1B0 (ranges as on left); Allow "\(x\)" |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.797 > 0.5577\) or \(-0.797 < -0.5577\) or \( | -0.797 | > 0.5577\) |
| Answer | Marks | Guidance |
|---|---|---|
| There are clusters (or groups etc.) | B1* | NOT: Too scattered; Not represent whole pop; Small sample |
| Apparent good correlation caused by clusters or two clusters with no \(-\)ve corr'n within them or a comment similar to one of the above. AND Conclusion: unreliable or Value of \(r\) is misleading oe | B1 dep B1* [2] | or Not bivariate normal distribution B1; so use of tables for \(r\) not valid B1; Clusters not on reg line B1B0 |
| Answer | Marks | Guidance |
|---|---|---|
| High prop of 65+ or Low prop of 18–24; Prop of young very similar, or \(\approx 0.06\); Proportion of senior to young is high | B1 [1] | If consider only one age-group, must be proportion not number; If consider both age-groups, allow e.g. Higher number of seniors than young or Many seniors, few young; NOT: Similar proportions of 65+; Population is elderly |
| Answer | Marks | Guidance |
|---|---|---|
| Top left points contain high prop of 18–24s. (So these LAs may be areas where there are universities or where they can recruit) | B1 [1] | Shows places where large nos of 18–24s; Shows where to focus recruiting; So universities can recruit; 18–24s are their target group; No need to specify "Top left group"; Allow "students" or "young" instead of "18–24s"; Any implication that diagram enables you to see information about location of young people |
# Question 11(a):
| $k > 1.4$ (allow $k > 1.1$ to $1.6$); $k < 0.25$ (allow $k < 0.2$ to $0.3$) | B1, B1 [2] | Allow $\geq$ and $\leq$; SC: $0.25 < k < 1.4$: B1B0 (ranges as on left); Allow "$x$" |
# Question 11(b)(i):
| $0.797 > 0.5577$ or $-0.797 < -0.5577$ or $|-0.797| > 0.5577$ | B2 [2] | $0.797 > 0.6055$ or $-0.797 < -0.6055$ B1; $\pm 0.5577$ B1; Allow $\geq$ or $\leq$ |
# Question 11(b)(ii):
| There are clusters (or groups etc.) | B1* | NOT: Too scattered; Not represent whole pop; Small sample |
| Apparent good correlation caused by clusters or two clusters with no $-$ve corr'n within them or a comment similar to one of the above. AND Conclusion: unreliable or Value of $r$ is misleading oe | B1 dep B1* [2] | or Not bivariate normal distribution B1; so use of tables for $r$ not valid B1; Clusters not on reg line B1B0 |
# Question 11(c):
| High prop of 65+ or Low prop of 18–24; Prop of young very similar, or $\approx 0.06$; Proportion of senior to young is high | B1 [1] | If consider only one age-group, must be proportion not number; If consider both age-groups, allow e.g. Higher number of seniors than young or Many seniors, few young; NOT: Similar proportions of 65+; Population is elderly |
# Question 11(d):
| Top left points contain high prop of 18–24s. (So these LAs may be areas where there are universities or where they can recruit) | B1 [1] | Shows places where large nos of 18–24s; Shows where to focus recruiting; So universities can recruit; 18–24s are their target group; No need to specify "Top left group"; Allow "students" or "young" instead of "18–24s"; Any implication that diagram enables you to see information about location of young people |
---11 A trainer was asked to give a lecture on population profiles in different Local Authorities (LAs) in the UK. Using data from the 2011 census, he created the following scatter diagram for 17 selected LAs.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{17 Selected Local Authorities}
\includegraphics[alt={},max width=\textwidth]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-08_560_897_466_246}
\end{center}
\end{figure}
He selected the 17 LAs using the following method. The proportions of people aged 18 to 24 and aged 65+ in any Local Authority are denoted by $P _ { \text {young } }$ and $P _ { \text {senior } }$ respectively. The trainer used a spreadsheet to calculate the value of $k = \frac { P _ { \text {young } } } { P _ { \text {senior } } }$ for each of the 348 LAs in the UK. He then used specific ranges of values of $k$ to select the 17 LAs.
\begin{enumerate}[label=(\alph*)]
\item Estimate the ranges of values of $k$ that he used to select these 17 LAs.
\item Using the 17 LAs the trainer carried out a hypothesis test with the following hypotheses.\\
$\mathrm { H } _ { 0 }$ : There is no linear correlation in the population between $P _ { \text {young } }$ and $P _ { \text {senior } }$.\\
$\mathrm { H } _ { 1 }$ : There is negative linear correlation in the population between $P _ { \text {young } }$ and $P _ { \text {senior } }$.\\
He found that the value of Pearson's product-moment correlation coefficient for the 17 LAs is - 0.797 , correct to 3 significant figures.
\begin{enumerate}[label=(\roman*)]
\item Use the table on page 9 to show that this value is significant at the $1 \%$ level.
The trainer concluded that there is evidence of negative linear correlation between $P _ { \text {young } }$ and $P _ { \text {senior } }$ in the population.
\item Use the diagram to comment on the reliability of this conclusion.
\end{enumerate}\item Describe one outstanding feature of the population in the areas represented by the points in the bottom right hand corner of the diagram.
\item The trainer's audience included representatives from several universities.
Suggest a reason why the diagram might be of particular interest to these people.
\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Critical values of Pearson's product-moment correlation coefficient}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multirow{2}{*}{1-tail test 2-tail test} & 5\% & 2.5\% & 1\% & 0.5\% \\
\hline
& 10\% & 5\% & 2\% & 1\% \\
\hline
$n$ & \multicolumn{4}{|c|}{} \\
\hline
1 & - & - & - & - \\
\hline
2 & - & - & - & - \\
\hline
3 & 0.9877 & 0.9969 & 0.9995 & 0.9999 \\
\hline
4 & 0.9000 & 0.9500 & 0.9800 & 0.9900 \\
\hline
5 & 0.8054 & 0.8783 & 0.9343 & 0.9587 \\
\hline
6 & 0.7293 & 0.8114 & 0.8822 & 0.9172 \\
\hline
7 & 0.6694 & 0.7545 & 0.8329 & 0.8745 \\
\hline
8 & 0.6215 & 0.7067 & 0.7887 & 0.8343 \\
\hline
9 & 0.5822 & 0.6664 & 0.7498 & 0.7977 \\
\hline
10 & 0.5494 & 0.6319 & 0.7155 & 0.7646 \\
\hline
11 & 0.5214 & 0.6021 & 0.6851 & 0.7348 \\
\hline
12 & 0.4973 & 0.5760 & 0.6581 & 0.7079 \\
\hline
13 & 0.4762 & 0.5529 & 0.6339 & 0.6835 \\
\hline
14 & 0.4575 & 0.5324 & 0.6120 & 0.6614 \\
\hline
15 & 0.4409 & 0.5140 & 0.5923 & 0.6411 \\
\hline
16 & 0.4259 & 0.4973 & 0.5742 & 0.6226 \\
\hline
17 & 0.4124 & 0.4821 & 0.5577 & 0.6055 \\
\hline
18 & 0.4000 & 0.4683 & 0.5425 & 0.5897 \\
\hline
19 & 0.3887 & 0.4555 & 0.5285 & 0.5751 \\
\hline
20 & 0.3783 & 0.4438 & 0.5155 & 0.5614 \\
\hline
21 & 0.3687 & 0.4329 & 0.5034 & 0.5487 \\
\hline
22 & 0.3598 & 0.4227 & 0.4921 & 0.5368 \\
\hline
23 & 0.3515 & 0.4132 & 0.4815 & 0.5256 \\
\hline
24 & 0.3438 & 0.4044 & 0.4716 & 0.5151 \\
\hline
25 & 0.3365 & 0.3961 & 0.4622 & 0.5052 \\
\hline
26 & 0.3297 & 0.3882 & 0.4534 & 0.4958 \\
\hline
27 & 0.3233 & 0.3809 & 0.4451 & 0.4869 \\
\hline
28 & 0.3172 & 0.3739 & 0.4372 & 0.4785 \\
\hline
29 & 0.3115 & 0.3673 & 0.4297 & 0.4705 \\
\hline
30 & 0.3061 & 0.3610 & 0.4226 & 0.4629 \\
\hline
\end{tabular}
\end{center}
\end{table}
Turn over for questions 12 and 13
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2019 Q11 [8]}}