Question 3 - A-Level Maths

OCR MEI Further Statistics Minor 2024 June — Question 3 13 marks

Exam Board	OCR MEI
Module	Further Statistics Minor (Further Statistics Minor)
Year	2024
Session	June
Marks	13
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Hypothesis test of Pearson’s product-moment correlation coefficient
Type	One-tailed test for positive correlation
Difficulty	Standard +0.3 This is a straightforward application of standard hypothesis testing for correlation with clearly provided summary statistics. While it's a Further Maths topic (making it slightly above average), the question requires only routine calculations of PMCC using the formula, stating hypotheses, and comparing to critical values. No novel insight or complex problem-solving is needed—just methodical application of learned procedures.
Spec	5.08a Pearson correlation: calculate pmcc 5.08c Pearson: measure of straight-line fit 5.08d Hypothesis test: Pearson correlation

3 The scatter diagram below illustrates data concerning average annual income per person, $\$ x$, and average life expectancy, $y$ years, for 45 randomly selected cities. \includegraphics[max width=\textwidth, alt={}, center]{464c80be-007b-4d5a-9fe5-2f35100bdea6-3_860_1465_354_244}

State whether neither variable, one variable or both variables can be considered to be random in this situation. A student is researching possible positive association between average annual income and average life expectancy. The student decides that the data point labelled A on the scatter diagram is an outlier.
Describe the apparent relationship between average annual income and average life expectancy for this data point relative to the rest of the data. The data for point A is removed. The student now wishes to carry out a hypothesis test using the product moment correlation coefficient for the remaining 44 data points to investigate whether there is positive correlation between average annual income and average life expectancy.
Explain why this type of hypothesis test is appropriate in this situation. Justify your answer. The summary statistics for these 44 data points are as follows. $\sum x = 751120 \sum y = 2397.1 \sum x ^ { 2 } = 14363849200 \sum y ^ { 2 } = 133014.63 \sum x y = 42465962$
Determine the value of the product moment correlation coefficient.
Carry out the test at the 1\% significance level.

Show mark scheme Show mark scheme source

Question 3:

Answer	Marks	Guidance
3	(a)	Both (variables)
[1]	2.2a
3	(b)	Compared to the rest of the data, A seems to have

a lower average life expectancy than might be

expected for its average annual income (or a

higher average annual income than might be

Answer	Marks	Guidance
expected for its average life expectancy)	B1
[1]	2.4
3	(c)	Such a hypothesis test is appropriate because the

data is random on random...

...and the data are consistent with an underlying

bivariate normal distribution since the points on

the scatter diagram seem to be elliptically

Answer	Marks
distributed	B1

B1

Answer	Marks
[2]	2.3
2.4	Allow ‘data is random’

Allow the data appear to be bivariate

normal

Not ‘normal bivariate’

Answer	Marks	Guidance
3	(d)	S = 14363849200 – 44*(751120/44)2 oe soi

xx

S = 133014.63 – 44*(2397.1/44)2 oe soi

yy

S = 42465962 – 44(751120/44) (2397.1/44) oe

xy

soi

S = awrt 1.54109

xx

S = awrt 2420

yy

S = awrt 1.55106

xy

Answer	Marks
r = awrt 0.80	M1

M1

B1

A1

Answer	Marks
[4]	1.1

1.1

Answer	Marks
1.1	Either S or S correctly written or

xx yy

value correctly found. Seen or used in

calculation.

Other values are possible e.g.

14363849200/44 – (751120/44)2

or 44*14363849200 –(751120)2

Seen or used in calculation.

Other values are possible e.g.

42465962/44 – (751120/44)

*(2397.1/44)

or 4442465962 –(751120) (2397.1)

but must be consistent throughout

question

For any one value correct. soi

Can’t be awarded without both M1

Answer	Marks
marks	1541547964

2421.711591...

1545285.818...

Answer	Marks	Guidance
3	(e)	𝐻 :𝜌 = 0

0 𝐻 :𝜌 > 0

1 Where 𝜌 is the population pmcc between average

annual income and average life expectancy

n = 44, 1-tailed critical value is 0.3496

0.800 > 0.3496 so we reject H

0 There is sufficient evidence (at the 1% level) to

suggest that there is positive correlation between

average annual income and average life

Answer	Marks
expectancy (in the population)	B1

B1

M1 FT

A1

Answer	Marks
[5]	3.3

2.5

3.4

1.1

Answer	Marks
2.2b	For both hypotheses

Use of r for 𝜌 is B0

For defining 𝜌 in context. Must

refer to ‘population.’

Could use x and y instead of income

and life expectancy.

Use of r for 𝜌 is B0

Condone 0.3457 (for n = 45)

Making correct comparison with

their values and reaching consistent

inference

0 < r < 1 required

Correct, non-assertive conclusion

from correct values only. Must

refer to context for this mark.

A0 if subsequent assertive

Answer	Marks
conclusion	H : There is no correlation between x

0 and y in the population

or

H : There is no correlation between

0 annual income and average life

expectancy in the population

H : There is positive correlation

1 between x and y in the population

or

H : There is positive correlation

1 between annual income and average

life expectancy in the population

If no reference to ‘population’ in at

least one of H and H B0

0 1

Question 3:
3 | (a) | Both (variables) | B1
[1] | 2.2a
3 | (b) | Compared to the rest of the data, A seems to have
a lower average life expectancy than might be
expected for its average annual income (or a
higher average annual income than might be
expected for its average life expectancy) | B1
[1] | 2.4
3 | (c) | Such a hypothesis test is appropriate because the
data is random on random...
...and the data are consistent with an underlying
bivariate normal distribution since the points on
the scatter diagram seem to be elliptically
distributed | B1
B1
[2] | 2.3
2.4 | Allow ‘data is random’
Allow the data appear to be bivariate
normal
Not ‘normal bivariate’
3 | (d) | S = 14363849200 – 44*(751120/44)2 oe soi
xx
S = 133014.63 – 44*(2397.1/44)2 oe soi
yy
S = 42465962 – 44*(751120/44) *(2397.1/44) oe
xy
soi
S = awrt 1.54109
xx
S = awrt 2420
yy
S = awrt 1.55106
xy
r = awrt 0.80 | M1
M1
B1
A1
[4] | 1.1
1.1
1.1
1.1 | Either S or S correctly written or
xx yy
value correctly found. Seen or used in
calculation.
Other values are possible e.g.
14363849200/44 – (751120/44)2
or 44*14363849200 –(751120)2
Seen or used in calculation.
Other values are possible e.g.
42465962/44 – (751120/44)
*(2397.1/44)
or 44*42465962 –(751120) *(2397.1)
but must be consistent throughout
question
For any one value correct. soi
Can’t be awarded without both M1
marks | 1541547964
2421.711591...
1545285.818...
3 | (e) | 𝐻 :𝜌 = 0
0
𝐻 :𝜌 > 0
1
Where 𝜌 is the population pmcc between average
annual income and average life expectancy
n = 44, 1-tailed critical value is 0.3496
0.800 > 0.3496 so we reject H
0
There is sufficient evidence (at the 1% level) to
suggest that there is positive correlation between
average annual income and average life
expectancy (in the population) | B1
B1
B1
M1 FT
A1
[5] | 3.3
2.5
3.4
1.1
2.2b | For both hypotheses
Use of r for 𝜌 is B0
For defining 𝜌 in context. Must
refer to ‘population.’
Could use x and y instead of income
and life expectancy.
Use of r for 𝜌 is B0
Condone 0.3457 (for n = 45)
Making correct comparison with
their values and reaching consistent
inference
0 < r < 1 required
Correct, non-assertive conclusion
from correct values only. Must
refer to context for this mark.
A0 if subsequent assertive
conclusion | H : There is no correlation between x
0
and y in the population
or
H : There is no correlation between
0
annual income and average life
expectancy in the population
H : There is positive correlation
1
between x and y in the population
or
H : There is positive correlation
1
between annual income and average
life expectancy in the population
If no reference to ‘population’ in at
least one of H and H B0
0 1

Show LaTeX source

3 The scatter diagram below illustrates data concerning average annual income per person, $\$ x$, and average life expectancy, $y$ years, for 45 randomly selected cities.\\
\includegraphics[max width=\textwidth, alt={}, center]{464c80be-007b-4d5a-9fe5-2f35100bdea6-3_860_1465_354_244}
\begin{enumerate}[label=(\alph*)]
\item State whether neither variable, one variable or both variables can be considered to be random in this situation.

A student is researching possible positive association between average annual income and average life expectancy. The student decides that the data point labelled A on the scatter diagram is an outlier.
\item Describe the apparent relationship between average annual income and average life expectancy for this data point relative to the rest of the data.

The data for point A is removed. The student now wishes to carry out a hypothesis test using the product moment correlation coefficient for the remaining 44 data points to investigate whether there is positive correlation between average annual income and average life expectancy.
\item Explain why this type of hypothesis test is appropriate in this situation. Justify your answer.

The summary statistics for these 44 data points are as follows.\\
$\sum x = 751120 \sum y = 2397.1 \sum x ^ { 2 } = 14363849200 \sum y ^ { 2 } = 133014.63 \sum x y = 42465962$
\item Determine the value of the product moment correlation coefficient.
\item Carry out the test at the 1\% significance level.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Statistics Minor 2024 Q3 [13]}}

This paper (6 questions)

View full paper

Q1 7 Q2 7 Q3 13 Q4 12 Q5 12 Q6 9