| Exam Board | WJEC |
|---|---|
| Module | Further Unit 2 (Further Unit 2) |
| Year | 2018 |
| 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 standard Spearman's rank correlation coefficient question with straightforward ranking, calculation using a formula (likely provided), and a one-tailed hypothesis test using critical value tables. The only mild challenge is the data handling with 7 values and checking for tied ranks, but the method is entirely routine for Further Maths statistics. Part (c) requires minimal interpretation. Slightly easier than average A-level due to its procedural nature. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Cow | A | B | C | D | E | F | G |
| Actual weight, kg | 614 | 1105 | 718 | 1001 | 889 | 770 | 682 |
| Estimated weight, kg | 700 | 1500 | 850 | 1400 | 750 | 900 | 800 |
| Answer | Marks | Guidance |
|---|---|---|
| Cow | A | B |
| Actual weight | 7 | 1 |
| Estimated weight | 7 | 1 |
| B1 B1 B1 | Correct values for first row. Correct values for second row. Accept reverse ranks. | |
| \(\sum d^2 = 12\) | ||
| \(r_s = 1 - \frac{6 \times 12}{7 \times 48}\) | M1 | |
| \(= 0.785(7142857\ldots)\) awrt 0.786 OR \(\frac{11}{14}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0\): There is no association between the actual weight and estimated weights of the cows. | B1 | Do not allow correlation. |
| Answer | Marks | Guidance |
|---|---|---|
| 5% 1-tail critical value = 0.6786 | B1 | |
| Reject \(H_0\) This suggests there is a positive association between the actual and estimated weights. | E1 | Either "Reject \(H_0\)" or "Positive association" FT their \(r_s\) |
| Answer | Marks | Guidance |
|---|---|---|
| It only shows they were good at putting the cows in weight order. The contestant may have been a long way out with their guesses. | E1 | B1 Anything which implies that this only shows they can order the cows. |
| [9] |
## 4(a)
The ranks are
| Cow | A | B | C | D | E | F | G |
|---|---|---|---|---|---|---|---|
| Actual weight | 7 | 1 | 5 | 2 | 3 | 4 | 6 |
| Estimated weight | 7 | 1 | 4 | 2 | 6 | 3 | 5 |
| B1 B1 B1 | Correct values for first row. Correct values for second row. Accept reverse ranks.
$\sum d^2 = 12$ | | |
$r_s = 1 - \frac{6 \times 12}{7 \times 48}$ | M1 |
$= 0.785(7142857\ldots)$ awrt 0.786 OR $\frac{11}{14}$ | A1 |
## 4(b)
$H_0$: There is no association between the actual weight and estimated weights of the cows. | B1 | Do not allow correlation.
$H_1$: There is a positive association between the actual weight and estimated weights of the cows.
5% 1-tail critical value = 0.6786 | B1 |
Reject $H_0$ This suggests there is a positive association between the actual and estimated weights. | E1 | Either "Reject $H_0$" or "Positive association" FT their $r_s$
## 4(c)
It only shows they were good at putting the cows in weight order. The contestant may have been a long way out with their guesses. | E1 | B1 Anything which implies that this only shows they can order the cows.
| [9] |
---On a Welsh television game show, contestants are asked to guess the weights of a random sample of seven cows. The game show judges want to investigate whether there is positive correlation between the actual weights and the estimated weights. The results are shown below for one contestant.
\begin{tabular}{|l|c|c|c|c|c|c|c|}
\hline
Cow & A & B & C & D & E & F & G \\
\hline
Actual weight, kg & 614 & 1105 & 718 & 1001 & 889 & 770 & 682 \\
\hline
Estimated weight, kg & 700 & 1500 & 850 & 1400 & 750 & 900 & 800 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\item Calculate Spearman's rank correlation coefficient for this data set. [5]
\item Stating your hypotheses clearly, determine whether or not there is evidence at the 5% significance level of a positive association between the actual weights and the weights as estimated by this contestant. [3]
\item One of the game show judges says, "This contestant was good at guessing the weights of the cows." Comment on this statement. [1]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 2 2018 Q4 [9]}}