| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2023 |
| Session | January |
| Marks | 12 |
| 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 standard hypothesis testing procedures for Spearman's rank correlation coefficient. Students must rank data, calculate rs using the formula, state hypotheses, and compare to critical values from tables. Part (c) involves a simple PMCC calculation from given summary statistics, and part (d) is another routine hypothesis test. All steps are procedural with no novel insight required, making it slightly easier than average. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Athlete | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) |
| Season's best time | 19.89 | 19.83 | 19.74 | 19.84 | 19.91 | 19.99 | 20.13 | 20.10 |
| Finishing position | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Ranks SBT: 4, 2, 1, 3, 5, 6, 8, 7; FP: 1, 2, 3, 4, 5, 6, 7, 8 | M1 | Attempt to rank seasonal best time (at least four correct); may be implied by \(\sum d^2 = 16\) |
| \(\sum d^2 = 9 + 0 + 4 + 1 + 0 + 0 + 1 + 1 = 16\) | M1 | Attempt to find the difference between each of the ranks (at least 3 correct) and evaluating \(\sum d^2\); may be implied by awrt 0.81; NB if no ranks for SBT it is M0 |
| \(r_s = 1 - \frac{6(\text{"16"})}{8(63)} = 0.8095...\) awrt 0.81 | dM1 A1 | Dependent on 1st M1; using \(1 - \frac{6\sum d^2}{8(63)}\) with their \(\sum d^2\); answer \(\frac{17}{21}\) or awrt 0.81(0); SC for reverse rankings may score M1M1dM1A0, order 5 7 8 6 4 3 1 2, \(\sum d^2 = 158\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \rho = 0\), \(H_1: \rho > 0\) | B1 | Both hypotheses correct; must be in terms of \(\rho\) (allow something that looks like rho e.g. \(p\)); must be attached to \(H_0\) and \(H_1\) |
| Critical value \(r_s = 0.8333\) or CR: \(r_s \geq 0.8333\) | B1 | Critical value of 0.8333; sign should match their \(H_1\) or \(r_s\); SC for two-tailed test: may score B0B1M1A0, CV allow 0.881... |
| Do not reject \(H_0\) / not significant / does not lie in the critical region / there is no evidence of a positive correlation | M1 | Correct statement comparing their CV with their \(r_s\); no context needed but do not allow contradicting non-contextual comments; if no CV or test statistic given or the \( |
| There is no evidence of a positive correlation between season's best time and finishing position for these athletes | A1ft | Correct conclusion in context for their value of \(r_s\) from (a) and their stated CV; conclusion must refer to positive correlation, seasonal best or time and position |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(r = \frac{0.225175}{\sqrt{0.1286875 \times 0.55275}} = 0.84428...\) awrt 0.844 | M1 A1 | Correct method used; awrt 0.844 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Critical value \(r = 0.7887\) or CR: \(r \geq 0.7887\) | M1 | Critical value of 0.7887; allow 0.8343 if hypotheses are two-tailed in (b) |
| So there is evidence of a positive correlation between season's best time and finishing time for these athletes | A1ft | M1 must be awarded; correct conclusion for their value of \(r\) from (c); conclusion must refer to positive correlation, seasonal best or time and finishing time; do not allow contradicting comments; if \( |
# Question 2:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Ranks SBT: 4, 2, 1, 3, 5, 6, 8, 7; FP: 1, 2, 3, 4, 5, 6, 7, 8 | M1 | Attempt to rank seasonal best time (at least four correct); may be implied by $\sum d^2 = 16$ |
| $\sum d^2 = 9 + 0 + 4 + 1 + 0 + 0 + 1 + 1 = 16$ | M1 | Attempt to find the difference between each of the **ranks** (at least 3 correct) and evaluating $\sum d^2$; may be implied by awrt 0.81; NB if no ranks for SBT it is M0 |
| $r_s = 1 - \frac{6(\text{"16"})}{8(63)} = 0.8095...$ awrt 0.81 | dM1 A1 | Dependent on 1st M1; using $1 - \frac{6\sum d^2}{8(63)}$ with their $\sum d^2$; answer $\frac{17}{21}$ or awrt 0.81(0); SC for reverse rankings may score M1M1dM1A0, order 5 7 8 6 4 3 1 2, $\sum d^2 = 158$ |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \rho = 0$, $H_1: \rho > 0$ | B1 | Both hypotheses correct; must be in terms of $\rho$ (allow something that looks like rho e.g. $p$); must be attached to $H_0$ and $H_1$ |
| Critical value $r_s = 0.8333$ or CR: $r_s \geq 0.8333$ | B1 | Critical value of 0.8333; sign should match their $H_1$ or $r_s$; SC for two-tailed test: may score B0B1M1A0, CV allow 0.881... |
| Do not reject $H_0$ / not significant / does not lie in the critical region / there is no evidence of a positive correlation | M1 | Correct statement comparing their CV with their $r_s$; no context needed but do not allow contradicting non-contextual comments; if no CV or test statistic given or the $|$test value$|$ or $|CV| > 1$ then it is M0 |
| There is no evidence of a **positive correlation** between **season's best time** and **finishing position** for these athletes | A1ft | Correct conclusion in context for their value of $r_s$ from (a) and their stated CV; conclusion must refer to **positive correlation**, **seasonal best** or **time** and **position** |
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $r = \frac{0.225175}{\sqrt{0.1286875 \times 0.55275}} = 0.84428...$ awrt 0.844 | M1 A1 | Correct method used; awrt 0.844 |
## Part (d)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Critical value $r = 0.7887$ or CR: $r \geq 0.7887$ | M1 | Critical value of 0.7887; allow 0.8343 if hypotheses are two-tailed in (b) |
| So there is evidence of a **positive correlation** between **season's best time** and **finishing time** for these athletes | A1ft | M1 must be awarded; correct conclusion for their value of $r$ from (c); conclusion must refer to **positive correlation**, **seasonal best** or **time** and **finishing time**; do not allow contradicting comments; if $|$test value$|$ or $|CV| > 1$ then it is M0 |
---2 The table shows the season's best times, $x$ seconds, for the 8 athletes who took part in the 200 m final in the 2021 Tokyo Olympics. It also shows their finishing position in the race.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | }
\hline
Athlete & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ \\
\hline
Season's best time & 19.89 & 19.83 & 19.74 & 19.84 & 19.91 & 19.99 & 20.13 & 20.10 \\
\hline
Finishing position & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
\end{tabular}
\end{center}
Given that the fastest season's best time is ranked number 1
\begin{enumerate}[label=(\alph*)]
\item calculate the value of the 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 rank of the season's best time and the finishing position for these athletes.
Chris suggests that it would be better to use the actual finishing time, $y$ seconds, of these athletes rather than their finishing position.
Given that
$$S _ { x x } = 0.1286875 \quad S _ { y y } = 0.55275 \quad S _ { x y } = 0.225175$$
\item calculate the product moment correlation coefficient between the season's best time and the finishing time for these athletes.\\
Give your answer correct to 3 decimal places.
\item Use your value of the product moment correlation coefficient to test, at the $1 \%$ level of significance, whether or not there is evidence of a positive correlation between the season's best time and the finishing time for these athletes.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2023 Q2 [12]}}