Question 16 - A-Level Maths

AQA Paper 3 2019 June — Question 16 10 marks

Exam Board	AQA
Module	Paper 3 (Paper 3)
Year	2019
Session	June
Marks	10
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Z-tests (known variance)
Type	Two-tail z-test
Difficulty	Moderate -0.3 This is a straightforward A-level statistics question testing standard concepts: (a) requires knowledge of the Large Data Set and interpretation (basic recall), (b) is a routine one-sample z-test with all values provided, and (c) tests understanding of Type I errors. All parts are textbook exercises with no novel problem-solving required, making it slightly easier than average.
Spec	2.02i Select/critique data presentation 2.05a Hypothesis testing language: null, alternative, p-value, significance 2.05e Hypothesis test for normal mean: known variance

The graph below shows the amount of salt, in grams, purchased per person per week in England between 2001–02 and 2014, based upon the Large Data Set. \includegraphics{figure_16a} Meera and Gemma are arguing about what this graph shows. Meera believes that the amount of salt consumed by people decreased greatly during this period. Gemma says that this is not the case. Using your knowledge of the Large Data Set, give two reasons why Gemma may be correct. [2 marks]
It is known that the mean amount of sugar purchased per person in England in 2014 was 78.9 grams, with a standard deviation of 25.0 grams. In 2018, a sample of 918 people had a mean of 80.4 grams of sugar purchased per person. Investigate, at the 5\% level of significance, whether the mean amount of sugar purchased per person in England has changed between 2014 and 2018. Assume that the survey data is a random sample taken from a normal distribution and that the standard deviation has remained the same. [6 marks]
Another test is performed to determine whether the mean amount of fat purchased per person has changed between 2014 and 2018. At the 10\% significance level, the null hypothesis is rejected. With reference to the 10\% significance level, explain why it is not necessarily true that there has been a change. [2 marks]

Show mark scheme Show mark scheme source

Question 16:

Answer	Marks
16(a)	States correct first reason involving

the y-axis

Accept there is no scale on the

Answer	Marks	Guidance
y-axis or graph does not start at 0	2.3	E1

zero.

Data is for salt purchased as

separate food stuff, not

consumed

States correct second reason

involving salt purchased and

consumed implying data not

Answer	Marks	Guidance
comparable	2.3	E1

Answer	Marks
16(b)	States both hypotheses correctly

for two-tailed test

Accept is

Answer	Marks	Guidance
78.9	2.5	B1

𝐻𝐻1:𝜇𝜇 ≠ 78.9

80.4−78.9

Test statistic=

25.0 �

√918

=Cr1iti.c8a2l z value 1.96

1.82 < 1.96

Accept - there is insufficient

evidence to suggest that the

am𝐻𝐻0

mean ount of sugar

purchased has changed

Formula𝐻𝐻te 0 s: 𝐩𝐩th𝐩𝐩e𝐩𝐩 t𝐩𝐩e𝐩𝐩s𝐩𝐩t𝐩𝐩 s𝐩𝐩𝐩𝐩ta𝐩𝐩t imsteica nor uses

the correct distribution of the

sample mean

PI by correct test statistic value or

probability or acceptance region

Condone

If region used, condone any z =

Answer	Marks	Guidance
(- 4, 4) 78.9−80.4	1.1a	M1

Obtains the correct value of the

test statistic 1.82 or

obtains the correct probability

0.0345 or 0.0691

obtains acceptance region of

Answer	Marks	Guidance
[77.3, 80.5]	1.1b	A1

Compares their 1.82 with 1.96 or

compares their 0.0345 with 0.025

compares their 0.0691 with 0.05 or

compares 80.4 with their region

Answer	Marks	Guidance
[77.3, 80.5]	1.1a	M1

Infers H accepted

0 CSO

Answer	Marks	Guidance
Must refer to	2.2b	A1

Correctly concludes in context that

there is insuf𝐻𝐻fic 0 ient evidence to

suggest that the mean amount of

sugar purchased has changed

Answer	Marks	Guidance
CSO	3.2a	E1

Answer	Marks
16(c)	Explains role of significance level in

rejecting null hypothesis in error

Answer	Marks	Guidance
Accept Type I error	2.3	E1

rejecting null hypothesis in error

Explains that there is 10 % chance

for this to occur

Reference to 10 % chance the

conclusion is incorrect scores

Answer	Marks	Guidance
E0E1	2.3	E1
Total	10
Q	Marking Instructions	AO

Question 16:
--- 16(a) ---
16(a) | States correct first reason involving
the y-axis
Accept there is no scale on the
y-axis or graph does not start at 0 | 2.3 | E1 | Scale on y-axis does not start at
zero.
Data is for salt purchased as
separate food stuff, not
consumed
States correct second reason
involving salt purchased and
consumed implying data not
comparable | 2.3 | E1
--- 16(b) ---
16(b) | States both hypotheses correctly
for two-tailed test
Accept is
78.9 | 2.5 | B1 | 𝐻𝐻 0:𝜇𝜇 = 78.9
𝐻𝐻1:𝜇𝜇 ≠ 78.9
80.4−78.9
Test statistic=
25.0
�
√918
=Cr1iti.c8a2l z value 1.96
1.82 < 1.96
Accept - there is insufficient
evidence to suggest that the
am𝐻𝐻0
mean ount of sugar
purchased has changed
Formula𝐻𝐻te 0 s: 𝐩𝐩th𝐩𝐩e𝐩𝐩 t𝐩𝐩e𝐩𝐩s𝐩𝐩t𝐩𝐩 s𝐩𝐩𝐩𝐩ta𝐩𝐩t imsteica nor uses
the correct distribution of the
sample mean
PI by correct test statistic value or
probability or acceptance region
Condone
If region used, condone any z =
(- 4, 4) 78.9−80.4 | 1.1a | M1
Obtains the correct value of the
test statistic 1.82 or
obtains the correct probability
0.0345 or 0.0691
obtains acceptance region of
[77.3, 80.5] | 1.1b | A1
Compares their 1.82 with 1.96 or
compares their 0.0345 with 0.025
compares their 0.0691 with 0.05 or
compares 80.4 with their region
[77.3, 80.5] | 1.1a | M1
Infers H accepted
0
CSO
Must refer to | 2.2b | A1
Correctly concludes in context that
there is insuf𝐻𝐻fic 0 ient evidence to
suggest that the mean amount of
sugar purchased has changed
CSO | 3.2a | E1
--- 16(c) ---
16(c) | Explains role of significance level in
rejecting null hypothesis in error
Accept Type I error | 2.3 | E1 | There is a 10% chance of
rejecting null hypothesis in error
Explains that there is 10 % chance
for this to occur
Reference to 10 % chance the
conclusion is incorrect scores
E0E1 | 2.3 | E1
Total | 10
Q | Marking Instructions | AO | Mark | Typical Solution

Show LaTeX source

\begin{enumerate}[label=(\alph*)]
\item The graph below shows the amount of salt, in grams, purchased per person per week in England between 2001–02 and 2014, based upon the Large Data Set.

\includegraphics{figure_16a}

Meera and Gemma are arguing about what this graph shows.

Meera believes that the amount of salt consumed by people decreased greatly during this period.

Gemma says that this is not the case.

Using your knowledge of the Large Data Set, give two reasons why Gemma may be correct.
[2 marks]

\item It is known that the mean amount of sugar purchased per person in England in 2014 was 78.9 grams, with a standard deviation of 25.0 grams.

In 2018, a sample of 918 people had a mean of 80.4 grams of sugar purchased per person.

Investigate, at the 5\% level of significance, whether the mean amount of sugar purchased per person in England has changed between 2014 and 2018.

Assume that the survey data is a random sample taken from a normal distribution and that the standard deviation has remained the same.
[6 marks]

\item Another test is performed to determine whether the mean amount of fat purchased per person has changed between 2014 and 2018.

At the 10\% significance level, the null hypothesis is rejected.

With reference to the 10\% significance level, explain why it is not necessarily true that there has been a change.
[2 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Paper 3 2019 Q16 [10]}}

This paper (17 questions)

View full paper

Q1 1 Q2 1 Q3 1 Q4 3 Q5 5 Q6 4 Q7 8 Q8 12 Q9 15 Q10 1 Q11 1 Q12 6 Q13 10 Q14 7 Q15 3 Q16 10 Q17 12