| Exam Board | Edexcel |
|---|---|
| Module | FS2 (Further Statistics 2) |
| Session | Specimen |
| 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 hypothesis test with the test statistic already provided. Part (a) requires standard hypothesis test procedure (stating H₀: ρ=0 vs H₁: ρ>0, comparing given rs=6/7 to critical value from tables). Parts (b) and (c) test conceptual understanding of how the coefficient changes under different scenarios and how to handle ties, but require no calculation. This is easier than average as it's a routine textbook exercise with the hardest computational work already done. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Position | A | B | C | D | E | F | G | ||
| 100 | 200 | 300 | 400 | 500 | 600 | 700 | ||
| Depth \(\boldsymbol { s } \mathbf { c m }\) | 60 | 75 | 85 | 76 | 110 | 120 | 104 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \rho = 0, H_1: \rho > 0\) | B1 | Both hypotheses correct written using the notation \(\rho\) |
| Critical value at 1% level is 0.8929 | B1 | awrt 0.893 |
| \(r_s < 0.8929\) so not significant evidence to reject \(H_0\) | M1 | |
| The researcher's claim is not correct (at 1% level) or insufficient evidence for researcher's claim or there is insufficient evidence that water gets deeper further from inner bank or no (positive) correlation between depth of water and distance from inner bank | A1ft |
| Answer | Marks | Guidance |
|---|---|---|
| The ranks will remain the same therefore there will be no change to the spearman's rank correlation coefficient | B1 | Stating no change and an explanation including ranks remain unchanged o.e., no change o.e. |
| Answer | Marks | Guidance |
|---|---|---|
| Spearman's rank correlation coefficient will increase since | B1 | Interpreted the outcome of adding a point as increased o.e |
| The ranks are the same for both distance and depth therefore \(d = 0\) however, \(n\) has increased or the new position follows the pattern that large \(b\) is associated with large \(s\) and so \(r_s\) will increase | B1 | Explaining why. Need to mention the ranks are the same for both o.e and \(n\) has increased o.e. |
| Answer | Marks |
|---|---|
| The mean of the tied ranks is given to each... | B1 |
| ...then use PMCC | B1 |
**(a)**
| $H_0: \rho = 0, H_1: \rho > 0$ | B1 | Both hypotheses correct written using the notation $\rho$ |
| Critical value at 1% level is 0.8929 | B1 | awrt 0.893 |
| $r_s < 0.8929$ so not significant evidence to reject $H_0$ | M1 | |
| The researcher's claim is not correct (at 1% level) **or** insufficient evidence for researcher's claim **or** there is insufficient evidence that water gets deeper further from inner bank **or** no (positive) correlation between depth of water and distance from inner bank | A1ft | |
**(b)(i)**
| The ranks will remain the same therefore there will be **no change** to the spearman's rank correlation coefficient | B1 | Stating **no change** and an explanation including ranks remain unchanged o.e., **no change** o.e. |
**(b)(ii)**
| Spearman's rank correlation coefficient will **increase** since | B1 | Interpreted the outcome of adding a point as **increased** o.e |
| The ranks are the same for both distance and depth therefore $d = 0$ however, $n$ has **increased** or the new position follows the pattern that large $b$ is associated with large $s$ and so $r_s$ will increase | B1 | Explaining why. Need to mention the ranks are the same for both o.e and $n$ has increased o.e. |
**(c)**
| The mean of the tied ranks is given to each... | B1 | |
| ...then use PMCC | B1 | |
---\begin{enumerate}
\item A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, $b \mathrm {~cm}$, and the depth of a river, $s \mathrm {~cm}$, at 7 positions. The results are shown in the table below.
\end{enumerate}
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | }
\hline
Position & A & B & C & D & E & F & G \\
\hline
\begin{tabular}{ l }
Distance from \\
inner bank $\boldsymbol { b } \mathbf { c m }$ \\
\end{tabular} & 100 & 200 & 300 & 400 & 500 & 600 & 700 \\
\hline
Depth $\boldsymbol { s } \mathbf { c m }$ & 60 & 75 & 85 & 76 & 110 & 120 & 104 \\
\hline
\end{tabular}
\end{center}
The Spearman's rank correlation coefficient between $b$ and $s$ is $\frac { 6 } { 7 }$\\
(a) Stating your hypotheses clearly, test whether or not the data provides support for the researcher's claim. Use a $1 \%$ level of significance.\\
(b) Without re-calculating the correlation coefficient, explain how the Spearman's rank correlation coefficient would change if\\
(i) the depth for G is 109 instead of 104\\
(ii) an extra value H with distance from the inner bank of 800 cm and depth 130 cm is included.
The researcher decided to collect extra data and found that there were now many tied ranks.\\
(c) Describe how you would find the correlation with many tied ranks.
\hfill \mbox{\textit{Edexcel FS2 Q2 [9]}}