| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics A AS (Further Statistics A AS) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| 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 application of a standard hypothesis test for correlation with n=10, using critical values from tables. Part (a) requires only stating hypotheses, comparing r=0.476 to a critical value at 10% significance, and concluding. While it's a Further Maths topic, the execution is routine with no problem-solving or novel insight required—slightly easier than an average A-level question due to its mechanical nature. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| Neuroticism | Extroversion | Openness | Agreeableness | Conscientiousness | |
| Neuroticism | 1 | ||||
| Extroversion | -0.296* | 1 | |||
| Openness | -0.044 | 0.405* | 1 | ||
| Agreeableness | -0.190* | 0.061 | 0.042 | 1 | |
| Conscientiousness | -0.485* | 0.145 | 0.235* | 0.112 | 1 |
| Effect size | ||
| 0.1 | Small | ||
| 0.3 | Medium | ||
| 0.5 | Large |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (a) | H : ρ = 0 |
| Answer | Marks |
|---|---|
| conscientiousness | B1 |
| Answer | Marks |
|---|---|
| [5] | 3.3 |
| Answer | Marks |
|---|---|
| 1.1 | Allow other symbols only if defined as population |
| Answer | Marks |
|---|---|
| (b) | For the second test the sample size was ‘much |
| Answer | Marks | Guidance |
|---|---|---|
| significant result with a lower test statistic. | B1 | |
| [1] | 2.2b | Accept reasonable alternatives. |
| (c) | Possible answers: |
| Answer | Marks |
|---|---|
| 5% level | B1 |
| Answer | Marks |
|---|---|
| [2] | 2.2b |
| 2.2b | For a suitable summary comment relating to effect size. |
Question 4:
4 | (a) | H : ρ = 0
0
H : ρ ≠ 0
1
Where ρ is the (population) correlation coefficient
between openness and conscientiousness
Critical value at 10% level is 0.5494
0.476 < 0.5494 so not significant
Insufficient evidence to suggest that there is
correlation between openness and
conscientiousness | B1
B1
B1
M1
A1
[5] | 3.3
2.5
3.4
2.2b
1.1 | Allow other symbols only if defined as population
correlation coefficient
For definition of ρ in context.
Condone use of 0.549
No further marks if incorrect c.v. used.
M1 for sensible comparison leading to a conclusion
For non-assertive correct conclusion in terms of H . See
1
additional notes.
(b) | For the second test the sample size was ‘much
larger’. This would lead to a smaller critical
value, even at 5% level, which could explain the
significant result with a lower test statistic. | B1
[1] | 2.2b | Accept reasonable alternatives.
(c) | Possible answers:
No correlations have large effect size.
Only 5 out 10 correlations are significant at the
5% level | B1
B1
[2] | 2.2b
2.2b | For a suitable summary comment relating to effect size.
e.g. All correlations involving Agreeableness have small
effect size.
For a further suitable summary comment relating to either
effect size or significance.
e.g. Three out of the four correlations involving
Neuroticism are found to be significant.4 A student is investigating correlations between various personality traits, two of which are conscientiousness and openness to new experiences.\\
She selects a random sample of 10 students at her university and uses standard tests to measure their conscientiousness and their openness. The product moment correlation coefficient between these two variables for the 10 students is 0.476 .
\begin{enumerate}[label=(\alph*)]
\item Assuming that the underlying population has a bivariate Normal distribution, carry out a hypothesis test at the $10 \%$ significance level to investigate whether there is any correlation between openness and conscientiousness in students.
Table 4.1 below shows the values of the product moment correlation coefficients between 5 different personality traits for a much larger sample of students. Those correlations that are significant at the $5 \%$ level are denoted by a * after the value of the correlation.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
& Neuroticism & Extroversion & Openness & Agreeableness & Conscientiousness \\
\hline
Neuroticism & 1 & & & & \\
\hline
Extroversion & -0.296* & 1 & & & \\
\hline
Openness & -0.044 & 0.405* & 1 & & \\
\hline
Agreeableness & -0.190* & 0.061 & 0.042 & 1 & \\
\hline
Conscientiousness & -0.485* & 0.145 & 0.235* & 0.112 & 1 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 4.1}
\end{center}
\end{table}
The student analyses these factors for effect size.\\
Guidelines often used when considering effect size are given in Table 4.2 below.
\begin{table}[h]
\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}
\captionsetup{labelformat=empty}
\caption{Table 4.2}
\end{center}
\end{table}
\item The student notes that, despite the result of the test in part (a), the correlation between openness and conscientiousness is significant at the $5 \%$ level with this second sample. Comment briefly on why this may be the case.
\item The student intends to summarise her findings about relationships between these factors, including effect sizes, in a report.\\
Use the information in Tables 4.1 and 4.2 to identify two summary points the student could make.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics A AS 2019 Q4 [8]}}