| Exam Board | OCR |
|---|---|
| Module | Further Statistics AS (Further Statistics AS) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| 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 straightforward application of hypothesis testing for correlation with standard critical value lookup (part a), followed by routine calculation of r using the formula (part b(i)), and basic interpretation (part b(ii)). All steps are standard textbook procedures requiring no novel insight, though the interpretation in (b)(ii) requires some statistical understanding of how restriction of range affects correlation. Slightly easier than average due to being mostly procedural. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \rho = 0,\ H_1: \rho \neq 0\), where \(\rho\) is population pmcc | B1 | Must use symbols. Allow no definition of letter if \(\rho\) used |
| \(0.701 > 0.4973\) | B1 | Correct CV stated, allow 0.497 |
| Reject \(H_0\). There is significant evidence of association between the marks on the two papers | M1ft A1 [4] | FT on wrong CV; *Not* FT. Needs context, and not too definite |
| Answer | Marks | Guidance |
|---|---|---|
| \(-0.534\) | B2 [2] | SC: if B0, give B1 for two of 1440, 2066, \(-921\) and \(S_{xy}/\sqrt{(S_{xx}S_{yy})}\) |
| Answer | Marks | Guidance |
|---|---|---|
| 6 candidates did very well or very badly on both papers; middle 10 tended to do badly on one paper and well on the other | B1 [1] | Correct inference about scores, *not* "correlation/association/value of \(r\)". *Not* "outliers" or "anomalies" |
# Question 5:
## Part (a)
$H_0: \rho = 0,\ H_1: \rho \neq 0$, where $\rho$ is population pmcc | **B1** | Must use symbols. Allow no definition of letter if $\rho$ used | *Not* "$H_0$: there is no assoc'n, $H_1$: there is association"
$0.701 > 0.4973$ | **B1** | Correct CV stated, allow 0.497
Reject $H_0$. There is significant evidence of association between the marks on the two papers | **M1ft A1 [4]** | FT on wrong CV; *Not* FT. Needs context, and not too definite | *Not* "There is association …"
## Part (b)(i)
$-0.534$ | **B2 [2]** | SC: if B0, give B1 for two of 1440, 2066, $-921$ and $S_{xy}/\sqrt{(S_{xx}S_{yy})}$
## Part (b)(ii)
6 candidates did very well or very badly on both papers; middle 10 tended to do badly on one paper and well on the other | **B1 [1]** | Correct inference about scores, *not* "correlation/association/value of $r$". *Not* "outliers" or "anomalies" | Allow inference for one group only, provided it *is* clearly for only one group & any ref to other group is not wrong
---5 Sixteen candidates took an examination paper in mechanics and an examination paper in statistics.
\begin{enumerate}[label=(\alph*)]
\item For all sixteen candidates, the value of the product moment correlation coefficient $r$ for the marks on the two papers was 0.701 correct to 3 significant figures.
Test whether there is evidence, at the $5 \%$ significance level, of association between the marks on the two papers.
\item A teacher decided to omit the marks of the candidates who were in the top three places in mechanics and the candidates who were in the bottom three places in mechanics. The marks for the remaining 10 candidates can be summarised by\\
$n = 10 , \sum x = 750 , \sum y = 690 , \sum x ^ { 2 } = 57690 , \sum y ^ { 2 } = 49676 , \sum x y = 50829$.
\begin{enumerate}[label=(\roman*)]
\item Calculate the value of $r$ for these 10 candidates.
\item What do the two values of $r$, in parts (a) and (b)(i), tell you about the scores of the sixteen candidates?
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics AS 2019 Q5 [7]}}