| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2021 |
| Session | November |
| Marks | 12 |
| 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 and standard procedure. Parts (b) and (c) test basic understanding of paired vs unpaired designs. Part (d) requires calculating combinations meeting a criterion, which is slightly more challenging but still routine for Further Statistics. Overall, this is slightly easier than average for A-level Further Maths statistics questions. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank5.07d Paired vs two-sample: selection |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (a) | H : Two samples are from identical populations |
| Answer | Marks |
|---|---|
| median ratings/opinions have changed | B2 |
| Answer | Marks |
|---|---|
| [8] | 1.1 |
| Answer | Marks |
|---|---|
| 2.2b | If no reference to “populations”, maximum B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (b) | Eliminate the difference between individual pupils’ opinions |
| [1] | 3.5b | “Minimises the difference in tastes” B1 (BOD) |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (c) | A paired-sample signed-rank test would have been used |
| [1] | signed rank” | |
| 7 | (d) | 0.025 × 12870 |
| = 322 | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.1a |
| 3.2a | 0.05 × 12870 = 643.5 M1 |
Question 7:
7 | (a) | H : Two samples are from identical populations
0
H : Two samples are from populations with different median
1
ratings.
R = 1 + 2 + 3 + 4 + 5 + 9 + 10 + 11 (= 45)
m
W = 45
8(8 + 8 + 1) – R = 91
m
W = 49
crit
Reject H . Significant evidence that there is a difference in
0
median ratings/opinions have changed | B2
M1
A1
B1
B1
M1ft
A1ft
[8] | 1.1
1.1
1.1
1.1
2.1
1.1
1.1
2.2b | If no reference to “populations”, maximum B1
Allow H : “identical population medians”, H : “not
0 1
identical populations” or “not identical pop medians”
“Pupils’ opinions have not changed”, etc: B2
If omitted, can still get all other marks
FT on TS (< 68) or CV
FT on TS only. Allow “increased”
SC: Sign or paired-sample test, max B2 (hypotheses)
7 | (b) | Eliminate the difference between individual pupils’ opinions | B1
[1] | 3.5b | “Minimises the difference in tastes” B1 (BOD)
Scores arbitrary: B1 (etc). Not “more powerful test”.
7 | (c) | A paired-sample signed-rank test would have been used | B1 | 3.5c | Must mention “paired sample” oe – not just “Wilcoxon
[1] | signed rank”
7 | (d) | 0.025 × 12870
= 322 | M1
A1
[2] | 3.1a
3.2a | 0.05 × 12870 = 643.5 M1
321 or 322 or 643 (from 1-tail), must be integer7 In a school opinion poll a random sample of 8 pupils were asked to rate school lunches on a scale of 0 to 20 . The results were as follows.\\
$\begin{array} { l l l l l l l l } 0 & 1 & 2 & 3 & 4 & 10 & 11 & 13 \end{array}$
After a new menu was introduced, the test was repeated with a different random sample of 8 pupils. The results were as follows.\\
$\begin{array} { l l l l l l l l } 7 & 8 & 9 & 14 & 15 & 17 & 19 & 20 \end{array}$
\begin{enumerate}[label=(\alph*)]
\item Carry out an appropriate Wilcoxon test at the $5 \%$ significance level to test whether pupils' opinions of school lunches have changed.
A statistics student tells the organisers of the opinion poll that it would have been better to have asked the same 8 pupils both times.
\item Explain why the statistics student's suggestion would produce a better test.
\item State which test should be used if the student's suggestion is followed.
\item You are given that there are 12870 ways in which 8 different integers can be chosen from the integers 1 to 16 inclusive.
Estimate the number of ways of selecting 8 different digits between 1 and 16 inclusive that have a sum less than or equal to the critical value used in the test in part (a).
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2021 Q7 [12]}}