| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| 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 data, standard hypothesis test setup, and n=5 making the calculation manageable. It requires ranking two variables, applying the formula, and comparing to critical values, but involves no conceptual subtleties or problem-solving beyond routine procedure. Slightly easier than average due to small dataset and direct application of a standard statistical test. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| \cline { 2 - 6 } \multicolumn{1}{c|}{} | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) |
| Coverage \% | 10 | 12 | 25 | 0 | 6 |
| Moisture \% | 30 | 20 | 40 | 10 | 25 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| Ranking table with columns \(A, B, C, D, E\): \(C\): 3, 2, 1, 5, 4; \(M\): 2, 4, 1, 5, 3; \(d\): 1, -2, 0, 0, 1; \(d^2\): 1, 4, 0, 0, 1 | M1 | Attempt to rank at least 1 row with at least 3 correct. Allow reverse rankings. |
| Attempt at \(d^2\) for their rankings | M1 | Can be implied by \(\sum d^2 = 6\) |
| \(\sum d^2 = 6\) | A1 | Can be implied by correct answer. Must come from correct rankings. |
| \(r_s = 1 - \dfrac{6(6)}{5(24)}\) | dM1 | Dependent on 1st M1. Use of correct formula with their \(\sum d^2\) |
| \(r_s = 0.7\) | A1 | 0.7 o.e. Must come from correct rankings. |
| (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(H_0: \rho = 0,\ H_1: \rho > 0\) | B1 | Both correct in terms of \(\rho\) or \(\rho_s\). Must be compatible with their ranking. |
| cv \(0.9\) or \(r_s \geq 0.9\) | B1 | Sign should match \(H_1\) or their \(r_s\) |
| \(r_s = 0.7\) does not lie in cr so do not reject \(H_0\) | M1 | Correct non-contextual statement e.g. "do not reject \(H_0\)", "not in critical region", "not significant", "no positive correlation". \( |
| Data does not support plant biologist's claim. | A1ft | Correct conclusion in context. Must mention "biologist's claim" o.e. or moisture and plant coverage. All previous marks in (b) must have been scored. |
| SC: For use of two-tailed test: May score B0B1M1A0 for cv = 1(.000) and 'not significant' oe | ||
| (4) | ||
| Total: 9 |
# Question 1:
## Part (a)
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Ranking table with columns $A, B, C, D, E$: $C$: 3, 2, 1, 5, 4; $M$: 2, 4, 1, 5, 3; $d$: 1, -2, 0, 0, 1; $d^2$: 1, 4, 0, 0, 1 | M1 | Attempt to rank at least 1 row with at least 3 correct. Allow reverse rankings. |
| Attempt at $d^2$ for their rankings | M1 | Can be implied by $\sum d^2 = 6$ |
| $\sum d^2 = 6$ | A1 | Can be implied by correct answer. Must come from correct rankings. |
| $r_s = 1 - \dfrac{6(6)}{5(24)}$ | dM1 | Dependent on 1st M1. Use of correct formula with their $\sum d^2$ |
| $r_s = 0.7$ | A1 | 0.7 o.e. Must come from correct rankings. |
| | | **(5)** |
## Part (b)
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $H_0: \rho = 0,\ H_1: \rho > 0$ | B1 | Both correct in terms of $\rho$ or $\rho_s$. Must be compatible with their ranking. |
| cv $0.9$ or $r_s \geq 0.9$ | B1 | Sign should match $H_1$ or their $r_s$ |
| $r_s = 0.7$ does not lie in cr so do not reject $H_0$ | M1 | Correct non-contextual statement e.g. "do not reject $H_0$", "not in critical region", "not significant", "no positive correlation". $|\text{test value}|$ or $|cv| > 1$ award M0 |
| Data does not support plant **biologist's claim**. | A1ft | Correct conclusion in context. Must mention "biologist's claim" o.e. or **moisture** and **plant coverage**. All previous marks in (b) must have been scored. |
| **SC**: For use of two-tailed test: May score B0B1M1A0 for cv = 1(.000) and 'not significant' oe | | |
| | | **(4)** |
| | | **Total: 9** |\begin{enumerate}
\item A plant biologist claims that as the percentage moisture content of the soil in a field increases, so does the percentage plant coverage. He splits the field into equal areas labelled $A , B , C , D$ and $E$ and measures the percentage plant coverage and the percentage moisture content for each area. The results are shown in the table below.
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\cline { 2 - 6 }
\multicolumn{1}{c|}{} & $A$ & $B$ & $C$ & $D$ & $E$ \\
\hline
Coverage \% & 10 & 12 & 25 & 0 & 6 \\
\hline
Moisture \% & 30 & 20 & 40 & 10 & 25 \\
\hline
\end{tabular}
\end{center}
(a) Calculate Spearman's rank correlation coefficient for these data.\\
(b) Stating your hypotheses clearly, test at the $5 \%$ level of significance, whether or not these data provide support for the plant biologist's claim.
\hfill \mbox{\textit{Edexcel S3 2021 Q1 [9]}}