| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Hypothesis test for positive correlation |
| Difficulty | Standard +0.8 This question requires understanding of Spearman's rank correlation hypothesis testing with critical values, working backwards from significance levels to find maximum Σd², and constructing a specific ranking satisfying constraints. While the calculations are straightforward once set up, the reverse-engineering in parts (b) and (c) requires problem-solving beyond routine application of the test procedure. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (a) | H : no association between orders in races of this |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | B1 | |
| [1] | 1.1 | Or clear equivalent, e.g. agreement, |
| Answer | Marks |
|---|---|
| Not “times” | Allow but not r or r |
| Answer | Marks |
|---|---|
| (b) | 6d2 |
| Answer | Marks |
|---|---|
| Largest possible value is 2 | M1 |
| Answer | Marks |
|---|---|
| [3] | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | Use correct formula and any tabular |
| Answer | Marks |
|---|---|
| = throughout | Allow > or = |
| Answer | Marks |
|---|---|
| (c) | d2 = 2 d = 0, 0, 0, 1, –1 in some order |
| Answer | Marks |
|---|---|
| (allow more than one correct answer) | M1ft |
| Answer | Marks |
|---|---|
| [3] | 3.1b |
| Answer | Marks |
|---|---|
| 2.2b | Turn d2 ≤ their 2 into possible |
| Answer | Marks |
|---|---|
| d2 = 3 but no other values | SC: d2 = 1, d= 00001 M1 |
Question 5:
5 | (a) | H : no association between orders in races of this
0
type; H : positive association between orders
1 | B1
[1] | 1.1 | Or clear equivalent, e.g. agreement,
independent, or H : = 0, H : > 0
0 s 1 s
Not “times” | Allow but not r or r
s
unless “population” explicit
(b) | 6d2
1 0.9
524
d 2 2
Largest possible value is 2 | M1
M1
A1
[3] | 3.1a
1.1a
1.1 | Use correct formula and any tabular
value of r (not 0.05 or from r)
s
Solve for d 2, needs correct formula
Correct conclusion, www but allow
= throughout | Allow > or =
then d 2 ≥ 2: M1M1A0
Allow from 0.05
Allow d2 < 2 so d2 = 1 if
working correct
(c) | d2 = 2 d = 0, 0, 0, 1, –1 in some order
E.g. ABCED or 12354
(allow more than one correct answer) | M1ft
M1ft
A1
[3] | 3.1b
2.2a
2.2b | Turn d2 ≤ their 2 into possible
values of d
Deduce possible order
Any order with two consecutive
runners interchanged, allow from
d2 = 3 but no other values | SC: d2 = 1, d= 00001 M1
1 explicitly impossible A1
Same order only A1
Else ignore ABCDE if seen
SC two-tailed (must be from “H : 0” only): B0; 1 – 6d2/120 1.0, d2 = 0 M1A1; “same order only” B1, total 3/7
1 s
SC wrong d2: in (c) allow M1M1 for showing any order giving d2 < their (b)5 Five runners, $A , B , C , D$ and $E$, take part in two different races.\\
Spearman's rank correlation coefficient for the orders in which the runners finish is calculated and a test for positive agreement is carried out at the $5 \%$ significance level.
\begin{enumerate}[label=(\alph*)]
\item State suitable hypotheses for the test.
\item Find the largest possible value of $\sum d ^ { 2 }$ for which the result of the test is to reject the null hypothesis.
\item In the first race, the order in which the five runners finished was: $A , B , C , D , E$. In the second race, three of the runners finished in the same positions as in the first race. The result of the test is to reject the null hypothesis.
Find a possible order for the runners to finish in the second race.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2019 Q5 [7]}}