Question 8 - A-Level Maths

OCR MEI Further Statistics Major 2022 June — Question 8 14 marks

Exam Board	OCR MEI
Module	Further Statistics Major (Further Statistics Major)
Year	2022
Session	June
Marks	14
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Hypothesis test of Spearman’s rank correlation coefficien
Type	Hypothesis test for association
Difficulty	Standard +0.3 This is a straightforward application of Spearman's rank correlation test with standard parts: explaining axis choice, justifying the test, performing the calculation (likely with given ranks or critical values), and commenting on interpretation. While it's a Further Maths topic, the question follows a routine structure with no novel problem-solving required, making it slightly easier than average overall.
Spec	2.02c Scatter diagrams and regression lines 2.02d Informal interpretation of correlation 2.05a Hypothesis testing language: null, alternative, p-value, significance 5.08e Spearman rank correlation 5.08f Hypothesis test: Spearman rank

8 A swimming coach is investigating whether there is correlation between the times taken by teenage swimmers to swim 50 m Butterfly and 50 m Freestyle. The coach selects a random sample of 11 teenage swimmers and records the times that each of them take for each event. The spreadsheet shows the data, together with a scatter diagram to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{77eabbd6-a058-457f-9601-d66f3c2db005-06_712_1465_456_274}

In the scatter diagram, Butterfly times have been plotted on the horizontal axis and Freestyle times on the vertical axis. A student states that the variables should have been plotted the other way around. Explain whether the student is correct. The student decides to carry out a hypothesis test to investigate whether there is any correlation between the times taken for the two events.
Explain why the student decides to carry out a test based on Spearman's rank correlation coefficient.
In this question you must show detailed reasoning. Carry out the test at the 5\% significance level.
The student concludes that there is definitely no correlation between the times. Comment on the student's conclusion.

Show mark scheme Show mark scheme source

Question 8:

Answer	Marks	Guidance
8	(a)	The other student is incorrect since both variables are

random

so it is equally correct to plot the variables either way

Answer	Marks
around.	E1

E1

Answer	Marks
[2]	2.2a
2.4	Condone ‘one variable does not affect the other’ or ‘both variables

are independent’

Dependent on first mark

Answer	Marks	Guidance
8	(b)	Because the scatter diagram does not appear to be

elliptical due to the outliers so the distribution is

Answer	Marks
probably not bivariate Normal.	E1

E1

Answer	Marks
[2]	3.5a
2.4	For elliptical

For full answer (dependent on first mark)

Condone 2 clusters instead of outliers

Answer	Marks	Guidance
8	(c)	DR

Rank B 1 2 3 4 5 6 7 8 9 10 11

Rank F 1 6 2 8 5 7 9 4 3 10 11

Spearman’s rank coefficient = 0.5909

H : There is no association between Butterfly and

0 Freestyle times in the population

H : There is some association between Butterfly and

1 Freestyle times in the population

For n = 11, 5% critical value is 0.6182

0.5909 < 0.6182

Do not reject H . There is insufficient evidence to

0 suggest that there is association between Butterfly and

Answer	Marks
Freestyle times [in the population]	M1

M1

A1

B1

M1

A1

Answer	Marks
[8]	1.1

1.1

3.3

1.2

3.4

1.1

Answer	Marks
2.2b	For ranking Butterfly

For ranking Freestyle Allow M1M1 for

may see 2

∑𝑑𝑑 = 90

6×90

1−11×(121−1)

BC Allow 0.59

For both. Do not allow correlation

Need to see population in one or other of the hypotheses for

second B1

Answer	Marks	Guidance
For comparison provided	r	< 1 (provided sensibly obtained) and

s

sensible critical value eg 0.6021

Do not FT their r Must be in context. Do not allow correlation

s

No marks for PMCC test

Answer	Marks	Guidance
8	(d)	If the test had been done at a different level, the

conclusion may have been different

Or A 5% significance level means that you will come

Answer	Marks
to the wrong conclusion 5% of the time.	E2
[2]	2.2b
2.4	Oe EG ‘With a different sample the results might be different’

Allow E1 for ‘The conclusion of a hypothesis test is never certain’

Answer	Marks	Guidance
Rank B	1	2
Rank F	1	6

Question 8:
8 | (a) | The other student is incorrect since both variables are
random
so it is equally correct to plot the variables either way
around. | E1
E1
[2] | 2.2a
2.4 | Condone ‘one variable does not affect the other’ or ‘both variables
are independent’
Dependent on first mark
8 | (b) | Because the scatter diagram does not appear to be
elliptical due to the outliers so the distribution is
probably not bivariate Normal. | E1
E1
[2] | 3.5a
2.4 | For elliptical
For full answer (dependent on first mark)
Condone 2 clusters instead of outliers
8 | (c) | DR
Rank B 1 2 3 4 5 6 7 8 9 10 11
Rank F 1 6 2 8 5 7 9 4 3 10 11
Spearman’s rank coefficient = 0.5909
H : There is no association between Butterfly and
0
Freestyle times in the population
H : There is some association between Butterfly and
1
Freestyle times in the population
For n = 11, 5% critical value is 0.6182
0.5909 < 0.6182
Do not reject H . There is insufficient evidence to
0
suggest that there is association between Butterfly and
Freestyle times [in the population] | M1
M1
A1
B1
B1
B1
M1
A1
[8] | 1.1
1.1
1.1
3.3
1.2
3.4
1.1
2.2b | For ranking Butterfly
For ranking Freestyle Allow M1M1 for
may see 2
∑𝑑𝑑 = 90
6×90
1−11×(121−1)
BC Allow 0.59
For both. Do not allow correlation
Need to see population in one or other of the hypotheses for
second B1
For comparison provided |r| < 1 (provided sensibly obtained) and
s
sensible critical value eg 0.6021
Do not FT their r Must be in context. Do not allow correlation
s
No marks for PMCC test
8 | (d) | If the test had been done at a different level, the
conclusion may have been different
Or A 5% significance level means that you will come
to the wrong conclusion 5% of the time. | E2
[2] | 2.2b
2.4 | Oe EG ‘With a different sample the results might be different’
Allow E1 for ‘The conclusion of a hypothesis test is never certain’
Rank B | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
Rank F | 1 | 6 | 2 | 8 | 5 | 7 | 9 | 4 | 3 | 10 | 11

Show LaTeX source

8 A swimming coach is investigating whether there is correlation between the times taken by teenage swimmers to swim 50 m Butterfly and 50 m Freestyle. The coach selects a random sample of 11 teenage swimmers and records the times that each of them take for each event. The spreadsheet shows the data, together with a scatter diagram to illustrate the data.\\
\includegraphics[max width=\textwidth, alt={}, center]{77eabbd6-a058-457f-9601-d66f3c2db005-06_712_1465_456_274}
\begin{enumerate}[label=(\alph*)]
\item In the scatter diagram, Butterfly times have been plotted on the horizontal axis and Freestyle times on the vertical axis. A student states that the variables should have been plotted the other way around.

Explain whether the student is correct.

The student decides to carry out a hypothesis test to investigate whether there is any correlation between the times taken for the two events.
\item Explain why the student decides to carry out a test based on Spearman's rank correlation coefficient.
\item In this question you must show detailed reasoning.

Carry out the test at the 5\% significance level.
\item The student concludes that there is definitely no correlation between the times.

Comment on the student's conclusion.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Statistics Major 2022 Q8 [14]}}

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