| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Statistics (Further Paper 3 Statistics) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Distribution |
| Type | Find threshold for given probability |
| Difficulty | Standard +0.3 This is a straightforward application of exponential distribution properties. Part (a) requires finding d from P(X<d)=0.05 using standard exponential CDF, then comparing with d=2. Part (b) asks for the pdf which is direct substitution into f(x)=λe^(-λx). Part (c) tests understanding that exponential distributions have mean=SD. All parts are routine recall and basic interpretation with no complex problem-solving, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{1}{\lambda} = 25 \Rightarrow \lambda = 0.04\) | M1 | Uses mean equalling 25 to find \(\lambda\) |
| \(P(X \leq 2) = 1 - e^{-0.04 \times 2}\) | M1 | Evaluates exponential model to find \(P(X \leq 2)\) or uses it to find \(x\) |
| \(= 0.077 > 0.05\) | A1 | Find \(P(X \leq 2)\) AWRT 0.077 or \(x\) AWRT 1.3 metres |
| The probability of a 'Red' rating has increased | E1F | Concludes probability higher or increased because 2 metres is higher than 1.3 metres; FT 'their' probability or \(x\) value |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(x) = 0.04e^{-0.04x}\), \(x \geq 0\) | B1F | States correct pdf for \(x \geq 0\) for their \(\lambda\) |
| \(f(x) = \begin{cases} 0.04e^{-0.04x} & x \geq 0 \\ 0 & \text{otherwise} \end{cases}\) | B1F | States complete correct pdf including \(x < 0\) (or otherwise) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\text{s.d.} = \sqrt{\frac{1}{0.04^2}} = 25\) metres | B1 | Obtains standard deviation of exponential model or value of \(\lambda(0.2)\) corresponding to s.d. of 5 metres |
| The model is not appropriate as the actual standard deviation of 5 metres is very different from the standard deviation of the model | E1 | Explains model not appropriate as actual s.d. not close to s.d. of model or \(\lambda\) values are different |
## Question 6:
### Part 6(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{\lambda} = 25 \Rightarrow \lambda = 0.04$ | M1 | Uses mean equalling 25 to find $\lambda$ |
| $P(X \leq 2) = 1 - e^{-0.04 \times 2}$ | M1 | Evaluates exponential model to find $P(X \leq 2)$ or uses it to find $x$ |
| $= 0.077 > 0.05$ | A1 | Find $P(X \leq 2)$ AWRT 0.077 or $x$ AWRT 1.3 metres |
| The probability of a 'Red' rating has increased | E1F | Concludes probability higher or increased because 2 metres is higher than 1.3 metres; FT 'their' probability or $x$ value |
### Part 6(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(x) = 0.04e^{-0.04x}$, $x \geq 0$ | B1F | States correct pdf for $x \geq 0$ for their $\lambda$ |
| $f(x) = \begin{cases} 0.04e^{-0.04x} & x \geq 0 \\ 0 & \text{otherwise} \end{cases}$ | B1F | States complete correct pdf including $x < 0$ (or otherwise) |
### Part 6(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\text{s.d.} = \sqrt{\frac{1}{0.04^2}} = 25$ metres | B1 | Obtains standard deviation of exponential model or value of $\lambda(0.2)$ corresponding to s.d. of 5 metres |
| The model is not appropriate as the actual standard deviation of 5 metres is very different from the standard deviation of the model | E1 | Explains model not appropriate as actual s.d. not close to s.d. of model or $\lambda$ values are different |
---6 The distance, $X$ metres, between successive breaks in a water pipe is modelled by an exponential distribution. The mean of $X$ is 25
The distance between two successive breaks is measured. A water pipe is given a 'Red' rating if the distance is less than $d$ metres.
The government has introduced a new law changing $d$ to 2\\
Before the government introduced the new law, the probability that a water pipe is given a 'Red' rating was 0.05
6
\begin{enumerate}[label=(\alph*)]
\item Explain whether or not the probability that a water pipe is given a 'Red' rating has increased as a result of the new law.\\
6
\item Find the probability density function of the random variable $X$.
6
\item After investigation, the distances between successive breaks in water pipes are found to have a standard deviation of 5 metres.
Explain whether or not the use of an exponential model in parts (a) and (b) is appropriate.\\[0pt]
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Statistics 2020 Q6 [8]}}