| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2023 |
| Session | June |
| Marks | 9 |
| 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 standard hypothesis testing procedures for correlation coefficients. Part (a) requires recall of textbook conditions, while parts (b) and (c) involve routine table lookups and comparison with critical values—no complex calculations or novel problem-solving required. Slightly above average difficulty only because it tests two different correlation coefficients and requires precise statement of hypotheses. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| When the data is ordinal e.g. Judges' ranks | B1 | For one correct condition |
| When a non-linear relationship might be expected | B1 | For a second correct condition. Condone not underlying normal |
| (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0: \rho = 0\), \(H_1: \rho \neq 0\) | B1 | For both hypotheses correct. Must be in terms of \(\rho\). Must be attached to \(H_0\) and \(H_1\) |
| Critical value \(r_s = -0.6485\) or CR: \(r_s \leq -0.6485\) (and \(r_s \geq 0.6485\)) | B1 | Allow \(-0.5636\) if one-tailed test stated for \(H_1\). Condone \(0.6485\) if compared with \(0.673\) |
| Reject \(H_0\) or significant or lies in the critical region | M1 | Correct statement; ft if CV is negative. Condone positive CV if comparison with \(0.673\) seen |
| The Spearman's rank correlation coefficient shows there is sufficient evidence of a correlation [between the length and maximum diameter of the melons] | A1 | For correct conclusion rejecting \(H_0\). Allow negative correlation. Independent of hypotheses |
| (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0: \rho = 0\), \(H_1: \rho < 0\) | B1 | For both hypotheses correct. Must be in terms of \(\rho\). Must be attached to \(H_0\) and \(H_1\) |
| Critical value \(r = -0.5494\) or CR: \(r \leq -0.5494\) | B1 | Allow \(-0.6319\) if two-tailed test stated for \(H_1\). Condone \(0.5494\) if compared with \(0.525\) |
| The product moment correlation coefficient shows there is insufficient evidence of a negative correlation [between the length and maximum diameter of the melons] | B1 | For a correct conclusion which is not rejecting \(H_0\) |
| (3) | ||
| Total 9 |
# Question 1:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| When the data is ordinal e.g. Judges' ranks | B1 | For one correct condition |
| When a non-linear relationship might be expected | B1 | For a second correct condition. Condone not underlying normal |
| | **(2)** | |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \rho = 0$, $H_1: \rho \neq 0$ | B1 | For both hypotheses correct. Must be in terms of $\rho$. Must be attached to $H_0$ and $H_1$ |
| Critical value $r_s = -0.6485$ or CR: $r_s \leq -0.6485$ (and $r_s \geq 0.6485$) | B1 | Allow $-0.5636$ if one-tailed test stated for $H_1$. Condone $0.6485$ if compared with $0.673$ |
| Reject $H_0$ or significant or lies in the critical region | M1 | Correct statement; ft if CV is negative. Condone positive CV if comparison with $0.673$ seen |
| The Spearman's rank correlation coefficient shows there is sufficient evidence of a correlation [between the length and maximum diameter of the melons] | A1 | For correct conclusion rejecting $H_0$. Allow negative correlation. Independent of hypotheses |
| | **(4)** | |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \rho = 0$, $H_1: \rho < 0$ | B1 | For both hypotheses correct. Must be in terms of $\rho$. Must be attached to $H_0$ and $H_1$ |
| Critical value $r = -0.5494$ or CR: $r \leq -0.5494$ | B1 | Allow $-0.6319$ if two-tailed test stated for $H_1$. Condone $0.5494$ if compared with $0.525$ |
| The product moment correlation coefficient shows there is insufficient evidence of a **negative** correlation [between the length and maximum diameter of the melons] | B1 | For a correct conclusion which is not rejecting $H_0$ |
| | **(3)** | |
| | **Total 9** | |\begin{enumerate}
\item (a) State two conditions under which it might be more appropriate to use Spearman's rank correlation coefficient rather than the product moment correlation coefficient.
\end{enumerate}
A random sample of 10 melons was taken from a market stall. The length, in centimetres, and maximum diameter, in centimetres, of each melon were recorded.
The Spearman's rank correlation coefficient between the results was - 0.673\\
(b) Test, at the $5 \%$ level of significance, whether or not there is evidence of a correlation. State clearly your hypotheses and the critical value used.
The product moment correlation coefficient between the results was - 0.525\\
(c) Test, at the $5 \%$ level of significance, whether or not there is evidence of a negative correlation.\\
State clearly your hypotheses and the critical value used.
\hfill \mbox{\textit{Edexcel S3 2023 Q1 [9]}}