| Exam Board | OCR |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2017 |
| Session | December |
| Marks | 9 |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Hypothesis test for association |
| Difficulty | Standard +0.3 This is a straightforward application of Spearman's rank correlation formula with standard hypothesis testing. Part (i) requires recognizing that Σd² has a maximum value of 70 for n=6, making 72 impossible. Parts (ii)-(iii) involve routine formula manipulation and table lookup. Part (iv) requires basic understanding that Spearman's and Pearson's tests measure similar things but may differ. All steps are procedural with no novel insight required, making this slightly easier than average. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank5.08g Compare: Pearson vs Spearman |
| Answer | Marks | Guidance |
|---|---|---|
| Gives \(-\frac{37}{35}\) which is \(< -1\) | M1 | Aim to establish inconsistency; Or: \(r = -1 \Rightarrow \max \sum d^2 = 70\) |
| A1 | Fully correct argument |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 - \frac{6\sum d^2}{6(6^2-1)} = \frac{29}{35}\) | M1 | Use formula and solve |
| \(\Rightarrow \sum d^2 = 6\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0\): No association between critics' rankings | B1 | |
| \(H_1\): Some association | B1 | |
| CV 0.886 | B1 | |
| \(0.829 < 0.886\) so do not reject \(H_0\) | M1 | Or 0.829 < 0.8857 |
| Insufficient evidence of association in the rankings | A1 | In context, acknowledge uncertainty; Not, e.g. "there is no association between the rankings" |
| Answer | Marks | Guidance |
|---|---|---|
| Not necessarily as testing different things (association/correlation) | B1 | Allow, e.g. "numerical ratings likely to be arbitrary" "even if raw data are not linearly correlated, ranks might be identical", etc |
## (i)
Gives $-\frac{37}{35}$ which is $< -1$ | M1 | Aim to establish inconsistency; Or: $r = -1 \Rightarrow \max \sum d^2 = 70$
| A1 | Fully correct argument
## (ii)
$1 - \frac{6\sum d^2}{6(6^2-1)} = \frac{29}{35}$ | M1 | Use formula and solve
$\Rightarrow \sum d^2 = 6$ | A1 |
## (iii)
$H_0$: No association between critics' rankings | B1 |
$H_1$: Some association | B1 |
CV 0.886 | B1 |
$0.829 < 0.886$ so do not reject $H_0$ | M1 | Or 0.829 < 0.8857
Insufficient evidence of association in the rankings | A1 | In context, acknowledge uncertainty; Not, e.g. "there is no association between the rankings"
## (iv)
Not necessarily as testing different things (association/correlation) | B1 | Allow, e.g. "numerical ratings likely to be arbitrary" "even if raw data are not linearly correlated, ranks might be identical", etc
---6 Arlosh, Sarah and Desi are investigating the ratings given to six different films by two critics.\\
(i) Arlosh calculates Spearman's rank correlation coefficient $r _ { s }$ for the critics' ratings. He calculates that $\Sigma d ^ { 2 } = 72$. Show that this value must be incorrect.\\
(ii) Arlosh checks his working with Sarah, whose answer $r _ { s } = \frac { 29 } { 35 }$ is correct. Find the correct value of $\Sigma d ^ { 2 }$.\\
(iii) Carry out an appropriate two-tailed significance test of the value of $r _ { s }$ at the $5 \%$ significance level, stating your hypotheses clearly.
Each critic gives a score out of 100 to each film. Desi uses these scores to calculate Pearson's product-moment correlation coefficient. She carries out a two-tailed significance test of this value at the $5 \%$ significance level.\\
(iv) Explain with a reason whether you would expect the conclusion of Desi's test to be the same as the result of the test in part (iii).
\hfill \mbox{\textit{OCR FS1 AS 2017 Q6 [9]}}