| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics A AS (Further Statistics A AS) |
| Year | 2024 |
| Session | June |
| Marks | 10 |
| 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 | Moderate -0.3 This is a straightforward hypothesis testing question requiring standard procedures: stating assumptions, comparing a test statistic to critical values from tables, and interpreting results. While it involves Further Maths content (correlation coefficient testing), the execution is routine with no novel problem-solving required. The multi-part structure and sample size comparison add some complexity but remain within standard textbook exercises. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| Effect size | ||
| 0.1 | Small | ||
| 0.3 | Medium | ||
| 0.5 | Large |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (a) | The underlying distribution is bivariate normal |
| [1] | 1.2 | Context not required. |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (b) | H : ρ = 0 |
| Answer | Marks |
|---|---|
| hours remaining awake (that day) | B1 |
| Answer | Marks |
|---|---|
| [5] | 3.3 |
| Answer | Marks |
|---|---|
| 3.5a | Allow hypotheses in words |
| Answer | Marks |
|---|---|
| correct CV and hypotheses only. | If one-tailed test used allow CV = |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (c) | (The new test statistic is 0.5487 and) the new CV |
| Answer | Marks |
|---|---|
| 0 | M1 |
| Answer | Marks |
|---|---|
| [2] | 2.2a |
| 2.2a | Identifying new CV |
| Answer | Marks |
|---|---|
| contextual conclusion here | If one-tailed test used allow CV = |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (d) | The second test is based on a larger sample so it |
| Answer | Marks |
|---|---|
| oe | B1 |
| Answer | Marks |
|---|---|
| [2] | 2.1 |
| 2.4 | This observation alone is sufficient |
| Answer | Marks |
|---|---|
| random factors | Allow “meaningful” for |
Question 5:
5 | (a) | The underlying distribution is bivariate normal | B1
[1] | 1.2 | Context not required. | Not “normal bivariate”.
Not “the data is bivariate normal”.
5 | (b) | H : ρ = 0
0
H : ρ ≠ 0
1
where ρ is the population correlation coefficient
between amount of coffee and number of waking
hours
(±) 0.7067
(–0.7067 <) 0.6030 < 0.7067
so we do not reject H oe
0
There is insufficient evidence at the 5% level to
suggest that there is any correlation between
volume of coffee drunk (in a day) and number of
hours remaining awake (that day) | B1
B1
B1
M1
A1
[5] | 3.3
2.4
3.4
2.2b
3.5a | Allow hypotheses in words
provided these refer to the
population correlation coefficient.
Defining ρ contextually.
CV for n = 8, 2-tailed test at 5%
For correctly comparing their test
statistic with their critical value
and then making a consistent
conclusion
For non-assertive conclusion in
context that refers to H from
1,
correct CV and hypotheses only. | If one-tailed test used allow CV =
0.6215 for B1FT
or “the result is not significant”
5 | (c) | (The new test statistic is 0.5487 and) the new CV
is 0.3610,
(0.5487 > 0.3610) so the result will be significant
oe e.g. H is rejected
0 | M1
A1
[2] | 2.2a
2.2a | Identifying new CV
Condone an assertive or non-
contextual conclusion here | If one-tailed test used allow CV =
0.3061 for M1. A1FT available.
5 | (d) | The second test is based on a larger sample so it
is likely to be more reliable.
The effect size is large so this test is informative
oe | B1
B1
[2] | 2.1
2.4 | This observation alone is sufficient
for B1.
The idea that the fact that the pmcc
is high enough for the test to be
significant is not just caused by
random factors | Allow “meaningful” for
“informative”.5 A student is investigating possible association between the amount of coffee that an adult drinks each day and the number of hours that they remain awake each day. In an initial investigation, a random sample of 8 adults is selected. The student obtains the following information from each of these adults: the amount of coffee that they drink each day and the number of hours that they remain awake each day.
The student analyses the data and finds that the associated product moment correlation coefficient is 0.6030 .
\begin{enumerate}[label=(\alph*)]
\item State one assumption that must be made for a hypothesis test based on the product moment correlation coefficient to be carried out.
For the remainder of this question you may assume that this assumption is true.
\item Carry out a test at the $5 \%$ significance level to investigate whether there is any correlation between amount of coffee drunk and number of hours awake.
The student conducts a second investigation which is similar to the first but this time based on a random sample of 30 adults. The product moment correlation coefficient for the new data is 0.5487 . The student carries out an equivalent hypothesis test to the one carried out in part (b), again using a 5\% significance level.
\item Identify any differences between the two tests and their results. You do not need to restate the hypotheses or explain the conclusion in context.
\item You may assume the following guidelines for considering effect size.
\begin{center}
\begin{tabular}{ | c | c | }
\hline
\begin{tabular}{ c }
Product moment \\
correlation coefficient \\
\end{tabular} & Effect size \\
\hline
0.1 & Small \\
\hline
0.3 & Medium \\
\hline
0.5 & Large \\
\hline
\end{tabular}
\end{center}
Explain briefly why the results of the student's second investigation are likely to be more reliable than the results of the initial investigation.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics A AS 2024 Q5 [10]}}