| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Statistics (Further Paper 3 Statistics) |
| Year | 2022 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Calculate simple probabilities |
| Difficulty | Moderate -0.8 This is a straightforward application of the rectangular (continuous uniform) distribution requiring only a single calculation: p = (12-10.5)/(12-8) = 0.375, then comparing to 0.4. Part (b) is basic interpretation. This is simpler than average A-level questions as it involves direct formula application with no problem-solving or multi-step reasoning. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)5.03a Continuous random variables: pdf and cdf |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X > 10.5) = \frac{1}{4} \times 1.5 = 0.375\) | B1 | Uses rectangular distribution model to obtain the correct value of \(P(X > 10.5)\); AO 3.4 |
| \(0.375 < 0.4\), Lianne will not buy the battery | E1F | Correctly compares their value of \(P(X > 10.5)\) with \(0.4\) and interprets the result in context; AO 3.2a |
| Answer | Marks | Guidance |
|---|---|---|
| The frequency density on the histogram is not approximately level between 8 and 12 hours | E1 | Recognises limitation of rectangular distribution in modelling situation with reference to the shape of the histogram; AO 3.5b |
| Use the normal distribution instead | B1 | Refines the model by suggesting the use of the normal distribution; AO 3.5c |
## Question 9(a):
$P(X > 10.5) = \frac{1}{4} \times 1.5 = 0.375$ | B1 | Uses rectangular distribution model to obtain the correct value of $P(X > 10.5)$; AO 3.4
$0.375 < 0.4$, Lianne will not buy the battery | E1F | Correctly compares their value of $P(X > 10.5)$ with $0.4$ and interprets the result in context; AO 3.2a
**Total: 2 marks**
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## Question 9(b):
The frequency density on the histogram is not approximately level between 8 and 12 hours | E1 | Recognises limitation of rectangular distribution in modelling situation with reference to the shape of the histogram; AO 3.5b
Use the normal distribution instead | B1 | Refines the model by suggesting the use of the normal distribution; AO 3.5c
**Total: 2 marks**
---
**Question total: 4 marks**
**Paper total: 50 marks**9 Lianne models the maximum time in hours that a rechargeable battery can be used, before needing to be recharged, with a rectangular distribution with values between 8 and 12
9
\begin{enumerate}[label=(\alph*)]
\item The probability that the maximum time the battery can be used before needing to be recharged is more than 10.5 hours is equal to $p$
Lianne will only buy the battery if $p$ is more than 0.4\\
Determine whether Lianne will buy the battery.\\[0pt]
[2 marks]\\
9
\item A histogram is plotted for 100 recharges showing the maximum time the battery can be used before needing to be recharged.\\
\includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-15_670_1186_404_427}
Explain why the model used in part (a) may not be valid and suggest the name of a different distribution that could be used to model the maximum time between recharges.\\
\includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-16_2488_1732_219_139}
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{62cee897-6eac-40b3-84c1-a0d165ba6903-20_2496_1721_214_148}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Statistics 2022 Q9 [4]}}