| Exam Board | AQA |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2018 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Mixed calculations with boundaries |
| Difficulty | Moderate -0.3 This is a routine statistics question testing standard calculations (mean, standard deviation) and basic normal distribution applications. Part (a) requires simple formula recall, parts (b) and (d) are textbook normal distribution problems, and part (c) tests understanding of model assumptions. All techniques are standard A-level content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.02g Calculate mean and standard deviation2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| 16(a)(i) | Obtains the correct mean | AO1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| 16(a)(ii) | Uses the correct formula for standard | |
| deviation | AO1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtains the correct standard deviation | AO1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 16(b)(i) | Uses the model to calculate a normal | |
| probability | AO3.4 | M1 |
| Obtains correct probability | AO1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 16(b)(ii) | Recalls correct value of 0 | AO1.2 |
| Answer | Marks | Guidance |
|---|---|---|
| 16(c) | Calculates the value of | |
| mean – 3 x standard deviations | AO1.1b | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 0 | AO2.2b | A1 |
| Answer | Marks |
|---|---|
| 16(d) | Standardises appropriately and |
| Answer | Marks | Guidance |
|---|---|---|
| PI by fully correct equation | AO3.1b | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| distribution (±1.2816) | AO1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| standardised result and z value | AO1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| value of μ | AO1.1b | A1 |
| Total | 12 | 𝜇𝜇 |
| Q | Marking Instructions | AO |
Question 16:
--- 16(a)(i) ---
16(a)(i) | Obtains the correct mean | AO1.1b | B1 | 1.38
--- 16(a)(ii) ---
16(a)(ii) | Uses the correct formula for standard
deviation | AO1.1a | M1 | 261.8 2
� −1.38
0.5 2 6 to 0.529
1 2 0
Obtains the correct standard deviation | AO1.1b | A1
--- 16(b)(i) ---
16(b)(i) | Uses the model to calculate a normal
probability | AO3.4 | M1 | 0.5417 to 0.5428
Obtains correct probability | AO1.1b | A1
--- 16(b)(ii) ---
16(b)(ii) | Recalls correct value of 0 | AO1.2 | B1 | 0
--- 16(c) ---
16(c) | Calculates the value of
mean – 3 x standard deviations | AO1.1b | M1 | -0.1998 to -0.207
This is less than 0 and so
model might not be
appropriate
Concludes that model might be
inappropriate as the value is less than
0 | AO2.2b | A1
--- 16(d) ---
16(d) | Standardises appropriately and
formalises a probability statement
PI by fully correct equation | AO3.1b | M1 | 0.75−𝜇𝜇
𝑃𝑃�𝑍𝑍 > � = 0.1
0.21
z value = 1.2816
0.75−𝜇𝜇
= 1.2816
0.21
=0.481
Obtains z value from inverse normal
distribution (±1.2816) | AO1.1a | M1
Forms a correct equation using
standardised result and z value | AO1.1b | A1
Solves the equation to find the correct
value of μ | AO1.1b | A1
Total | 12 | 𝜇𝜇
Q | Marking Instructions | AO | Marks | Typical SolutionA survey of 120 adults found that the volume, $X$ litres per person, of carbonated drinks they consumed in a week had the following results:
$$\sum x = 165.6 \quad \sum x^2 = 261.8$$
\begin{enumerate}[label=(\alph*)]
\item
\begin{enumerate}[label=(\roman*)]
\item Calculate the mean of $X$.
[1 mark]
\item Calculate the standard deviation of $X$.
[2 marks]
\end{enumerate}
\item Assuming that $X$ can be modelled by a normal distribution find
\begin{enumerate}[label=(\roman*)]
\item P$(0.5 < X < 1.5)$
[2 marks]
\item P$(X = 1)$
[1 mark]
\end{enumerate}
\item Determine with a reason, whether a normal distribution is suitable to model this data.
[2 marks]
\item It is known that the volume, $Y$ litres per person, of energy drinks consumed in a week may be modelled by a normal distribution with standard deviation 0.21
Given that P$(Y > 0.75) = 0.10$, find the value of $\mu$, correct to three significant figures.
[4 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 3 2018 Q16 [12]}}