| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Wilcoxon rank-sum test (Mann-Whitney U test) |
| Difficulty | Standard +0.3 This is a straightforward application of the Wilcoxon rank-sum test with clear data already ordered. Part (i) requires recognizing that unequal variances violate t-test assumptions (routine knowledge), and part (ii) involves standard ranking procedure, calculating the test statistic, and comparing to tables. The data is conveniently pre-sorted and sample sizes are small, making calculations mechanical rather than requiring problem-solving insight. |
| Spec | 5.07d Paired vs two-sample: selection |
| M | A | A | M | M | M | M | M | M | A | A | A | A | A | A |
| 19 | 20 | 22 | 24 | 25 | 26 | 28 | 30 | 31 | 33 | 35 | 37 | 38 | 39 | 42 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Differentiate to obtain form \(kx(2x^2 + 9)^n\) | M1 | any constant k; any n < \(\frac{5}{2}\) |
| Obtain correct \(10x(2x^2 + 9)^{\frac{1}{2}}\) | A1 | or (unsimplified) equiv |
| Equate to 100 and confirm \(x = 10(2x^2 + 9)^{-\frac{1}{2}}\) | A1 | AG; necessary detail required |
| (ii) Attempt relevant calculations with 0.3 and 0.4 | M1 | Obtain at least one correct value |
| \(x\) | \(f(x)\) | \(x - f(x)\) |
| 0.3 | 0.3595 | -0.0595 |
| 0.4 | 0.3515 | 0.0485 |
| A1 | Obtain two correct values and conclude appropriately | |
| A1 | noting sign change or showing 0.3 < f(0.3) and 0.4 > f(0.4) or showing gradients either side of 100 | |
| (iii) Obtain correct first iterate | B1 | |
| Carry out correct process | M1 | finding at least 3 iterates in all |
| Obtain 0.3553 | A1 | answer required to exactly 4 dp |
(i) Differentiate to obtain form $kx(2x^2 + 9)^n$ | M1 | any constant k; any n < $\frac{5}{2}$
Obtain correct $10x(2x^2 + 9)^{\frac{1}{2}}$ | A1 | or (unsimplified) equiv
Equate to 100 and confirm $x = 10(2x^2 + 9)^{-\frac{1}{2}}$ | A1 | AG; necessary detail required
(ii) Attempt relevant calculations with 0.3 and 0.4 | M1 | Obtain at least one correct value
| | $x$ | $f(x)$ | $x - f(x)$ | $f'(x)$
| | 0.3 | 0.3595 | -0.0595 | 83.4
| | 0.4 | 0.3515 | 0.0485 | 113.8
| A1 | Obtain two correct values and conclude appropriately
| A1 | noting sign change or showing 0.3 < f(0.3) and 0.4 > f(0.4) or showing gradients either side of 100
(iii) Obtain correct first iterate | B1 |
Carry out correct process | M1 | finding at least 3 iterates in all
Obtain 0.3553 | A1 | answer required to exactly 4 dp
[0.3 → 0.35953 → 0.35497 → 0.35534 → 0.35531;
0.35 → 0.35575 → 0.35528 → 0.35532 (→ 0.35531);
0.4 → 0.35146 → 0.35563 → 0.35529 → 0.35532]4 William takes a bus regularly on the same journey, sometimes in the morning and sometimes in the afternoon. He wishes to compare morning and afternoon journey times. He records the journey times on 7 randomly chosen mornings and 8 randomly chosen afternoons. The results, each correct to the nearest minute, are as follows, where M denotes a morning time and A denotes an afternoon time.
\begin{center}
\begin{tabular}{ c c c c c c c c c c c c c c c }
M & A & A & M & M & M & M & M & M & A & A & A & A & A & A \\
19 & 20 & 22 & 24 & 25 & 26 & 28 & 30 & 31 & 33 & 35 & 37 & 38 & 39 & 42 \\
\end{tabular}
\end{center}
William wishes to test for a difference between the average times of morning and afternoon journeys.\\
(i) Given that $s _ { M } ^ { 2 } = 16.5$ and $s _ { A } ^ { 2 } = 64.5$, with the usual notation, explain why a $t$-test is not appropriate in this case.\\
(ii) William chooses a non-parametric test at the $5 \%$ significance level. Carry out the test, stating the rejection region.
\hfill \mbox{\textit{OCR S4 2008 Q4 [7]}}