| Exam Board | OCR |
|---|---|
| Module | Further Statistics AS (Further Statistics AS) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Pearson’s product-moment correlation coefficient |
| Type | Calculate PMCC from summary statistics |
| Difficulty | Standard +0.3 This is a standard Further Statistics question testing routine application of PMCC formula and hypothesis testing procedures. Part (i) requires substituting into a formula booklet expression, parts (ii)-(iii) are textbook hypothesis test steps, and part (iv) tests understanding that linear transformations don't affect correlation. All parts follow predictable patterns with no novel problem-solving required, though it's slightly above average difficulty due to being Further Maths content. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.4(00)\) BC | B2 [2] | SC: if B0, give SC B1 for two of \(S_{xx} = 12500\), \(S_{yy} = 1600\), \(S_{xy} = 1790\) and \(S_{xy}/\sqrt{(S_{xx}S_{yy})}\). Also allow SC B1 for equivalent methods using Covariance & SDs |
| Answer | Marks | Guidance |
|---|---|---|
| Data needs to have a bivariate normal distribution | B1 [1] | Needs "bivariate normal" or clear equivalent. Not just "both normally distributed". Allow "scatter diagram forms ellipse" |
| Answer | Marks | Guidance |
|---|---|---|
| Significant evidence that higher maths scores are associated with higher BCF grading | B1 M1ft A1ft [4] | B1: Needs context and clearly one-tailed OR \(\rho\) used and defined. Not "evidence that…". \(H_0: \rho = 0\), \(H_1: \rho > 0\) where \(\rho\) is population pmcc (not \(r\)). B1: Allow 0.378. M1ft: Reject/do not reject \(H_0\), FT on their \(r\), but not CV. A1ft: Contextualised, not too definite. Needn't say "positive" if \(H_1\) OK. SC 2-tail: B0; 0.4438, or 0.3783 B1; then M1A0. Not "scores are associated…". FT on their \(r\) only |
| Answer | Marks | Guidance |
|---|---|---|
| It makes no difference as this is a linear transformation | B1 [1] | Need both "unchanged" and reason, need "linear" or exact equivalent. "oe" includes "their 0.4" |
## Question 4:
### Part (i)
$0.4(00)$ **BC** | **B2** [2] | SC: if B0, give SC B1 for two of $S_{xx} = 12500$, $S_{yy} = 1600$, $S_{xy} = 1790$ and $S_{xy}/\sqrt{(S_{xx}S_{yy})}$. Also allow SC B1 for equivalent methods using Covariance & SDs
### Part (ii)
Data needs to have a bivariate normal distribution | **B1** [1] | Needs "bivariate normal" or clear equivalent. Not just "both normally distributed". Allow "scatter diagram forms ellipse"
### Part (iii)
$H_0$: higher maths scores are not associated with higher BCF grading; $H_1$: positively associated
CV $0.3783$
$0.400 > 0.3783$ so reject $H_0$
Significant evidence that higher maths scores are associated with higher BCF grading | **B1** M1ft A1ft [4] | B1: Needs context and clearly one-tailed OR $\rho$ used and defined. Not "evidence that…". $H_0: \rho = 0$, $H_1: \rho > 0$ where $\rho$ is population pmcc (not $r$). B1: Allow 0.378. M1ft: Reject/do not reject $H_0$, FT on their $r$, but not CV. A1ft: Contextualised, not too definite. Needn't say "positive" if $H_1$ OK. SC 2-tail: B0; 0.4438, or 0.3783 B1; then M1A0. Not "scores are associated…". FT on their $r$ only
### Part (iv)
It makes no difference as this is a linear transformation | **B1** [1] | Need both "unchanged" and reason, need "linear" or exact equivalent. "oe" includes "their 0.4"
---4 Judith believes that mathematical ability and chess-playing ability are related. She asks 20 randomly chosen chess players, with known British Chess Federation (BCF) ratings $X$, to take a mathematics aptitude test, with scores $Y$. The results are summarised as follows.
$$n = 20 , \sum x = 3600 , \sum x ^ { 2 } = 660500 , \sum y = 1440 , \sum y ^ { 2 } = 105280 , \sum x y = 260990$$
(i) Calculate the value of Pearson's product-moment correlation coefficient $r$.\\
(ii) State an assumption needed to be able to carry out a significance test on the value of $r$.\\
(iii) Assume now that the assumption in part (ii) is valid. Test at the $5 \%$ significance level whether there is evidence that chess players with higher BCF ratings are better at mathematics.\\
(iv) There are two different grading systems for chess players, the BCF system and the international ELO system. The two sets of ratings are related by
$$\text { ELO rating } = 8 \times \text { BCF rating } + 650$$
Magnus says that the experiment should have used ELO ratings instead of BCF ratings. Comment on Magnus's suggestion.
\hfill \mbox{\textit{OCR Further Statistics AS 2018 Q4 [8]}}