| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2008 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Hypothesis test for positive correlation |
| Difficulty | Standard +0.3 This is a straightforward application of Spearman's rank correlation coefficient with clear rankings already provided. Part (a) tests basic understanding of correlation concepts through sketching, while part (b) involves standard calculation of r_s using the formula and a routine hypothesis test against critical values from tables. The question requires no novel insight—just methodical application of learned procedures. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Judge 1 | \(A\) | \(C\) | \(D\) | \(B\) | \(E\) | \(F\) | \(G\) |
| Judge 2 | \(A\) | \(B\) | \(D\) | \(C\) | \(E\) | \(G\) | \(F\) |
| Answer | Marks | Guidance |
|---|---|---|
| Scatter plot with 5 or more points on a straight line of positive gradient | B1 | (3 marks) |
| Answer | Marks |
|---|---|
| Scatter plot with 5 or more points showing pattern of negative correlation not on a straight line | B1B1 |
| Answer | Marks | Guidance |
|---|---|---|
| A | B | C |
| Rank (Judge 1) | 1 | 4 |
| Rank (Judge 2) | 1 | 2 |
| \(d^2\) | 0 | 4 |
| M1M1 | ||
| \(\sum d^2 = 10\) | M1A1 | |
| \(r_s = 1 - \frac{6 \times 10}{7 \times (49-1)} = 1 - \frac{5}{28} = \frac{23}{28}\) or awrt 0.821 | M1A1 | (6 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \rho = 0\) | B1 | |
| \(H_1: \rho > 0\) (Allow \(\rho_s\)) | B1 | (Allow \(H_1: \rho \neq 0\) scores B0) |
| \(r_s\) 5% one tail critical value is 0.7143 | B1 | |
| Significant result or reject null hypothesis | M1 | |
| There is evidence of a (positive) correlation between the judges or the judges agree | A1ft | (5 marks) |
**Part (a)(i)**
Scatter plot with 5 or more points on a straight line of positive gradient | B1 | (3 marks)
**Part (a)(ii)**
Scatter plot with 5 or more points showing pattern of negative correlation not on a straight line | B1B1 |
**Part (b)(i)**
| | **A** | **B** | **C** | **D** | **E** | **F** | **G** |
|---|---|---|---|---|---|---|---|
| **Rank (Judge 1)** | 1 | 4 | 2 | 3 | 5 | 6 | 7 |
| **Rank (Judge 2)** | 1 | 2 | 4 | 3 | 5 | 7 | 6 |
| **$d^2$** | 0 | 4 | 4 | 0 | 0 | 1 | 1 |
| M1M1 |
$\sum d^2 = 10$ | M1A1 |
$r_s = 1 - \frac{6 \times 10}{7 \times (49-1)} = 1 - \frac{5}{28} = \frac{23}{28}$ or awrt **0.821** | M1A1 | (6 marks)
**Part (b)(ii)**
$H_0: \rho = 0$ | B1 |
$H_1: \rho > 0$ (Allow $\rho_s$) | B1 | (Allow $H_1: \rho \neq 0$ scores B0)
$r_s$ 5% one tail critical value is **0.7143** | B1 |
Significant result or reject null hypothesis | M1 |
There is evidence of a (positive) correlation between the judges or the judges agree | A1ft | (5 marks)
**Guidance Notes:**
1st M1 for attempting to rank one of the judges (at least 2 correct rankings)
2nd M1 for ranking both (may be reversed) (at least 2 correct rankings)
3rd M1 for attempting $d^2$
1st A1 for $\sum d^2 = 10$
4th M1 for correct use of the $r_s$ formula
3rd B1 for the correct critical value - depends upon their $H_1$: $\rho > 0$ needs 0.7143, $\rho \neq 0$, 0.7857
The $H_1$ may be in words so B0B1 is possible. If no $H_1$ award for 0.7143 only.
5th M1 for a correct statement relating their $r_s$ and their cv (may be implied by correct comment)
3rd A1ft follow through their $r_s$ and their cv. Comment in context. Must mention judges.
Don't insist on "positive" and condone if they are using $\rho \neq 0$.
---\begin{enumerate}
\item The product moment correlation coefficient is denoted by $r$ and Spearman's rank correlation coefficient is denoted by $r _ { s }$.\\
(a) Sketch separate scatter diagrams, with five points on each diagram, to show\\
(i) $r = 1$,\\
(ii) $r _ { s } = - 1$ but $r > - 1$.
\end{enumerate}
Two judges rank seven collie dogs in a competition. The collie dogs are labelled $A$ to $G$ and the rankings are as follows
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | }
\hline
Rank & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
Judge 1 & $A$ & $C$ & $D$ & $B$ & $E$ & $F$ & $G$ \\
\hline
Judge 2 & $A$ & $B$ & $D$ & $C$ & $E$ & $G$ & $F$ \\
\hline
\end{tabular}
\end{center}
(b) (i) Calculate Spearman's rank correlation coefficient for these data.\\
(ii) Stating your hypotheses clearly, test, at the $5 \%$ level of significance, whether or not the judges are generally in agreement.
\hfill \mbox{\textit{Edexcel S3 2008 Q3 [14]}}