| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| 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 coefficient with standard hypothesis testing. Part (a) requires ranking and calculation using the formula (routine but computational), part (b) is a standard hypothesis test comparing to critical values from tables, and part (c) tests understanding of when to use Spearman's vs Pearson's (likely due to non-linear relationship visible in the data). While multi-part, each component is textbook standard with no novel insight required, making it slightly easier than average. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| 1.1 | 1.3 | 1.6 | 2.1 | 2.4 | 2.6 | 2.8 | 3.0 | ||
| Sales | 527 | 632 | 840 | 619 | 350 | 425 | 487 | 401 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | capacity | 1.1 |
| sales | 527 | 632 |
| cap. rank | 8 | 7 |
| sales rank | 4 | 2 |
| \(d^2\) | 16 | 25 |
| \(\sum d^2 = 140\) | M2, A2 | |
| \(r_s = 1 - \frac{6d40}{8×63} = -0.6667\) | M1, A1 | |
| (b) \(H_0: \rho = 0\) \(H_1: \rho \neq 0\) | B1 | |
| \(n = 8\), 5% level ∴ C.R. is \(r_s < -0.7381\) or \(r_s > 0.7381\) not in C.R. ∴ no evidence of correlation | M1, A1, A1 | |
| (c) need variables to be jointly normally distributed for pmcc test engine capacities are discrete so use Spearman's | B2 | (12) |
**(a)** | capacity | 1.1 | 1.3 | 1.6 | 2.1 | 2.4 | 2.6 | 2.8 | 3.0 |
|---|---|---|---|---|---|---|---|---|
| sales | 527 | 632 | 840 | 619 | 350 | 425 | 487 | 401 |
| cap. rank | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
| sales rank | 4 | 2 | 1 | 3 | 8 | 6 | 5 | 7 |
| $d^2$ | 16 | 25 | 25 | 4 | 16 | 9 | 9 | 36 |
$\sum d^2 = 140$ | M2, A2 |
$r_s = 1 - \frac{6d40}{8×63} = -0.6667$ | M1, A1 |
**(b)** $H_0: \rho = 0$ $H_1: \rho \neq 0$ | B1 |
$n = 8$, 5% level ∴ C.R. is $r_s < -0.7381$ or $r_s > 0.7381$ not in C.R. ∴ no evidence of correlation | M1, A1, A1 |
**(c)** need variables to be jointly normally distributed for pmcc test engine capacities are discrete so use Spearman's | B2 | (12) |
---4. For a project a student collects data on engine size and sales over a period of time for the models of cars made by one particular manufacturer. Her results are shown in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | }
\hline
\begin{tabular}{ c }
Engine Capacity \\
(litres) \\
\end{tabular} & 1.1 & 1.3 & 1.6 & 2.1 & 2.4 & 2.6 & 2.8 & 3.0 \\
\hline
Sales & 527 & 632 & 840 & 619 & 350 & 425 & 487 & 401 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate Spearman's rank correlation coefficient for these data.
\item Stating your hypotheses clearly, test at the $5 \%$ level of significance whether or not there is any evidence of correlation.
\item Explain why it is more appropriate to use Spearman's rank correlation coefficient for this test than the product moment correlation coefficient.\\
(2 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 Q4 [12]}}