| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2022 |
| Session | January |
| Marks | 8 |
| 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 requiring ranking of qualifying times, calculation using the standard formula, and a one-tailed hypothesis test against critical values. While it involves multiple steps, each is routine for S3 students with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Driver | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) | ||
| 62.94 | 63.92 | 63.63 | 62.95 | 63.97 | 63.87 | 64.31 | 64.64 | 65.18 | 64.21 | ||
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Ranking FQL: A=1, B=5, C=3, D=2, E=6, F=4, G=8, H=9, I=10, J=7 | M1 | Attempt to rank fastest qualifying lap (at least four correct) |
| \(\sum d^2 = 0+9+0+4+1+4+1+1+1+9 = 30\) | M1 | Finding the difference between each of the ranks and evaluating \(\sum d^2\) |
| \(r_s = 1 - \frac{6(30)}{10(99)}\) | dM1 | Dependent on 1st M1. Using \(1 - \frac{6\sum d^2}{10(99)}\) with their \(\sum d^2\) |
| \(= 0.8181818...\) awrt 0.818 | A1 | \(\frac{9}{11}\) or awrt 0.818 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \rho = 0\), \(H_1: \rho > 0\) | B1 | Both hypotheses correct. Must be in terms of \(\rho\). Must be attached to \(H_0\) and \(H_1\) |
| Critical Value \(r_s = 0.7455\) or CR: \(r_s \geq 0.7455\) | B1 | Critical value of 0.7455 |
| Reject \(H_0\) or significant or lies in the critical region | M1 | A correct statement comparing their CV with their \(r_s\) - no context needed but do not allow contradicting non-contextual comments |
| There is sufficient evidence of a positive correlation between fastest qualifying lap time and finishing position for these Formula One racing drivers | A1 | Correct conclusion which is rejecting \(H_0\), which must mention lap time and finishing position |
## Question 3:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Ranking FQL: A=1, B=5, C=3, D=2, E=6, F=4, G=8, H=9, I=10, J=7 | M1 | Attempt to rank fastest qualifying lap (at least four correct) |
| $\sum d^2 = 0+9+0+4+1+4+1+1+1+9 = 30$ | M1 | Finding the difference between each of the ranks and evaluating $\sum d^2$ |
| $r_s = 1 - \frac{6(30)}{10(99)}$ | dM1 | Dependent on 1st M1. Using $1 - \frac{6\sum d^2}{10(99)}$ with their $\sum d^2$ |
| $= 0.8181818...$ awrt 0.818 | A1 | $\frac{9}{11}$ or awrt 0.818 |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \rho = 0$, $H_1: \rho > 0$ | B1 | Both hypotheses correct. Must be in terms of $\rho$. Must be attached to $H_0$ and $H_1$ |
| Critical Value $r_s = 0.7455$ or CR: $r_s \geq 0.7455$ | B1 | Critical value of 0.7455 |
| Reject $H_0$ or significant or lies in the critical region | M1 | A correct statement comparing their CV with their $r_s$ - no context needed but do not allow contradicting non-contextual comments |
| There is sufficient evidence of a positive correlation between fastest qualifying **lap time** and **finishing position** for these Formula One racing drivers | A1 | Correct conclusion which is rejecting $H_0$, which must mention **lap time** and **finishing position** |
---3. The table shows the time, in seconds, of the fastest qualifying lap for 10 different Formula One racing drivers and their finishing position in the actual race.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | }
\hline
Driver & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ & $I$ & $J$ \\
\hline
\begin{tabular}{ l }
Fastest \\
qualifying lap \\
\end{tabular} & 62.94 & 63.92 & 63.63 & 62.95 & 63.97 & 63.87 & 64.31 & 64.64 & 65.18 & 64.21 \\
\hline
\begin{tabular}{ l }
Finishing \\
position \\
\end{tabular} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate the value of Spearman's rank correlation coefficient for these data.
\item Stating your hypotheses clearly, test at the $1 \%$ level of significance, whether or not there is evidence of a positive correlation between the fastest qualifying lap time and finishing position for these Formula One racing drivers.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2022 Q3 [8]}}