| Exam Board | WJEC |
|---|---|
| Module | Further Unit 2 (Further Unit 2) |
| Year | 2023 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Given coefficient value only |
| Difficulty | Standard +0.3 This is a straightforward Further Maths statistics question testing standard knowledge of Spearman's rank correlation. Part (a) requires recall of when to use Spearman's vs Pearson's. Part (b)(i) is a routine hypothesis test with the test statistic given. Part (b)(ii) involves ranking data and applying the standard Spearman's formula—mechanical but with 6 data points. Part (b)(iii) requires basic interpretation. All parts are textbook-standard with no novel insight required, making it slightly easier than average even for Further Maths. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank5.08g Compare: Pearson vs Spearman |
| Tractor | Farmer's rank | PTO (horsepower) | Price (£1000s) |
| A | 1 | 77·5 | 80 |
| B | 6 | 87·9 | 45 |
| C | 5 | 53·0 | 47 |
| D | 4 | 41·0 | 53 |
| E | 2 | 112·0 | 60 |
| F | 3 | 90·0 | 61 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (a) Two appropriate circumstances. e.g. When the data are ordinal. e.g. When the data are not bivariate normal. | B2 | B1 for one circumstance. |
| (b)(i) \(H_0\): there is no association between the rank given by the farmer and the price. \(H_1\): there is an association between the rank given by the farmer and the price. | B1 | Both. Do not allow correlation. |
| 5% two tailed critical value \(= (\pm)0.8286\) | B1 | |
| Since 0.9429 > 0.8286 there is sufficient evidence to reject \(H_0\). | B1 | Comparing TS to CV; FT their CV |
| There is sufficient evidence to suggest that there is an association between the rank given by the farmer and price. | E1 | Comment in context. Only award if previous 3 B1 have been awarded |
| (ii) | ||
| Tractor table with columns: Farmer's rank, PTO rank (highest to lowest), PTO rank (lowest to highest) | B1 | Correct values for one set of PTO ranks |
| \(\sum d^2 = 24\) or \(\sum d^2 = 46\) | B1 | si |
| \(r_s = 1 - \frac{6 \times 24}{6 \times 35}\) or \(r_s = 1 - \frac{6 \times 46}{6 \times 35}\) | M1 | FT their '24' or '46' |
| \(r_s = \pm 0.3143\) or \(\pm \frac{11}{35}\) | A1 | |
| (iii) Two valid comment. e.g. The PTO variable isn't very strongly associated with the farmer's ranks so may not be worth analysing. e.g. The salesman doesn't need to work that hard because the farmer already prefers the most expensive tractors. e.g. because the number of tractors in the sample is small, the salesman should not place too much reliance on the results. | E2 | |
| Total [12] |
| Answer/Working | Mark | Guidance |
|---|---|---|
| **(a)** Two appropriate circumstances. e.g. When the data are ordinal. e.g. When the data are not bivariate normal. | B2 | B1 for one circumstance. |
| **(b)(i)** $H_0$: there is no association between the rank given by the farmer and the price. $H_1$: there is an association between the rank given by the farmer and the price. | B1 | Both. Do not allow correlation. |
| 5% two tailed critical value $= (\pm)0.8286$ | B1 | |
| Since 0.9429 > 0.8286 there is sufficient evidence to reject $H_0$. | B1 | Comparing TS to CV; FT their CV |
| There is sufficient evidence to suggest that there is an association between the rank given by the farmer and price. | E1 | Comment in context. Only award if previous 3 B1 have been awarded |
| **(ii)** | | |
| Tractor table with columns: Farmer's rank, PTO rank (highest to lowest), PTO rank (lowest to highest) | B1 | Correct values for one set of PTO ranks |
| $\sum d^2 = 24$ or $\sum d^2 = 46$ | B1 | si |
| $r_s = 1 - \frac{6 \times 24}{6 \times 35}$ or $r_s = 1 - \frac{6 \times 46}{6 \times 35}$ | M1 | FT their '24' or '46' |
| $r_s = \pm 0.3143$ or $\pm \frac{11}{35}$ | A1 | |
| **(iii)** Two valid comment. e.g. The PTO variable isn't very strongly associated with the farmer's ranks so may not be worth analysing. e.g. The salesman doesn't need to work that hard because the farmer already prefers the most expensive tractors. e.g. because the number of tractors in the sample is small, the salesman should not place too much reliance on the results. | E2 | |
| **Total [12]** | | |\begin{enumerate}[label=(\alph*)]
\item Give two circumstances where it may be more appropriate to use Spearman's rank correlation coefficient rather than Pearson's product moment correlation coefficient. [2]
\item A farmer needs a new tractor. The tractor salesman selects 6 tractors at random to show the farmer. The farmer ranks these tractors, in order of preference, according to their ability to meet his needs on the farm. The tractor salesman makes a note of the price and power take-off (PTO) of the tractors.
\begin{tabular}{|c|c|c|c|}
\hline
Tractor & Farmer's rank & PTO (horsepower) & Price (£1000s) \\
\hline
A & 1 & 77·5 & 80 \\
B & 6 & 87·9 & 45 \\
C & 5 & 53·0 & 47 \\
D & 4 & 41·0 & 53 \\
E & 2 & 112·0 & 60 \\
F & 3 & 90·0 & 61 \\
\hline
\end{tabular}
Spearman's rank correlation coefficient between the farmer's ranks and the price is 0·9429.
\begin{enumerate}[label=(\roman*)]
\item Test at the 5% significance level whether there is an association between the price of a tractor and the farmer's judgement of the ability of the tractor to meet his needs on the farm. [4]
\item Calculate Spearman's rank correlation coefficient between the farmer's rank and PTO. [4]
\item How should the tractor salesman interpret the results in (i) and (ii)? [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 2 2023 Q5 [12]}}