| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Minor (Further Statistics Minor) |
| Year | 2021 |
| Session | November |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Pearson’s product-moment correlation coefficient |
| Type | Two-tailed test for any correlation |
| Difficulty | Standard +0.3 This is a standard hypothesis testing question on PMCC requiring routine calculations and interpretation. Part (b) uses a formula from the data booklet, part (c) is a textbook two-tailed test procedure, and parts (a), (d), (e) require standard statistical commentary. While it's a Further Maths topic, the execution is mechanical with no novel problem-solving required. |
| Spec | 2.02c Scatter diagrams and regression lines2.02d Informal interpretation of correlation2.05a Hypothesis testing language: null, alternative, p-value, significance5.05c Hypothesis test: normal distribution for population mean5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (a) | Because (the grouping of points on) the scatter |
| Answer | Marks |
|---|---|
| the distribution may be bivariate Normal. | E1 |
| Answer | Marks |
|---|---|
| [2] | 3.5a |
| 2.4 | For elliptical |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (b) | S =12317.2− 1 ×351.9×350.0=0.7 |
| Answer | Marks |
|---|---|
| = 0.60 | M1 |
| Answer | Marks |
|---|---|
| [4] | 1.1a |
| Answer | Marks |
|---|---|
| 1.1 | For either S or S |
| Answer | Marks |
|---|---|
| intermediate calculations | Detailed reasoning |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (c) | H : ρ = 0 |
| Answer | Marks |
|---|---|
| directly measured and satellite measured salinity level. | B1 |
| Answer | Marks |
|---|---|
| [5] | 3.3 |
| Answer | Marks |
|---|---|
| 2.2b | For both hypotheses |
| Answer | Marks |
|---|---|
| FT for conclusion in context | H : no correlation in |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (d) | It means that one can be more confident that the |
| Answer | Marks | Guidance |
|---|---|---|
| random variation. | E1 | |
| [1] | 2.4 | |
| 4 | (e) | The test shows that there is almost certainly some real |
| Answer | Marks |
|---|---|
| effect size is so small. | E1 |
| Answer | Marks |
|---|---|
| [2] | 3.5a |
Question 4:
4 | (a) | Because (the grouping of points on) the scatter
diagram appears to be very roughly elliptical,
the distribution may be bivariate Normal. | E1
E1
[2] | 3.5a
2.4 | For elliptical
For full answer (dep. on first mark)
“the data is bivariate Normal” is E0
4 | (b) | S =12317.2− 1 ×351.9×350.0=0.7
xy 10
S =12384.5− 1 ×351.92 =1.139
xx 10
S =12251.2− 1 ×350.02 =1.2
yy 10
S 0.7
r = xy =
S S 1.139×1.2
xx yy
= 0.60 | M1
M1
M1
A1
[4] | 1.1a
1.1
3.3
1.1 | For either S or S
xx yy
For general form including square root
SC2 for correct value without any
intermediate calculations | Detailed reasoning
required.
(0.598750…)
4 | (c) | H : ρ = 0
0
H : ρ > 0
1
where ρ is the population product moment correlation
coefficient between x and y
For n = 10, 5% critical value (one tailed) = 0.5494
Since 0.60 > 0.5494 the result is significant.
There is sufficient evidence (at the 5% level) to
suggest that there is positive correlation between
directly measured and satellite measured salinity level. | B1
B1
B1
M1
A1FT
[5] | 3.3
2.5
3.4
1.1
2.2b | For both hypotheses
For defining ρ
For critical value
For comparison of test statistic and
critical value leading to a conclusion
FT for conclusion in context | H : no correlation in
0
the population
H : positive
1
correlation in the
population
scores first B1
4 | (d) | It means that one can be more confident that the
correlation is genuine, rather than simply the result of
random variation. | E1
[1] | 2.4
4 | (e) | The test shows that there is almost certainly some real
correlation in the population.
However, it is uninformative/of little use since the
effect size is so small. | E1
E1
[2] | 3.5a
2.2b4 A scientist is investigating sea salinity (the level of salt in the sea) in a particular area. She wishes to check whether satellite measurements, $y$, of salinity are similar to those directly measured, $x$. Both variables are measured in parts per thousand in suitable units.
The scientist obtains a random sample of 10 values of $x$ and the related values of $y$. Below is a screenshot of a scatter diagram to illustrate the data. She decides to carry out a hypothesis test to check if there is any correlation between direct measurement, $x$, and satellite measurement, $y$.\\
\includegraphics[max width=\textwidth, alt={}, center]{691e8b55-e9a1-4fff-b9ee-a71ff1f73ead-5_830_837_589_246}
\begin{enumerate}[label=(\alph*)]
\item Explain why the scientist might decide to carry out a test based on the product moment correlation coefficient.
Summary statistics for $x$ and $y$ are as follows.\\
$n = 10 \quad \sum x = 351.9 \quad \sum y = 350.0 \quad \sum x ^ { 2 } = 12384.5 \quad \sum y ^ { 2 } = 12251.2 \quad \sum \mathrm { xy } = 12317.2$
\item In this question you must show detailed reasoning.
Calculate the product moment correlation coefficient.
\item Carry out a hypothesis test at the $5 \%$ significance level to investigate whether there is positive correlation between directly measured and satellite measured salinity levels.
\item Explain why it would be preferable to use a larger sample.
The scientist is also interested in whether there is any correlation between salinity and numbers of a particular species of shrimp in the water. She takes a large sample and finds that the product moment correlation coefficient for this sample is 0.165 . The result of a test based on this sample is to reject the null hypothesis and conclude that there is correlation between salinity and numbers of shrimp.
\item Comment on the outcome of the hypothesis test with reference to the effect size of 0.165 .
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Minor 2021 Q4 [14]}}