| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Statistics (Further Paper 3 Statistics) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Find parameters from given statistics |
| Difficulty | Standard +0.3 This is a straightforward application of standard formulas for continuous uniform distributions. Part (a) requires recalling that for X~U(a,b), mean=(a+b)/2 and variance=(b-a)²/12, then solving two simultaneous equations. Part (b) involves basic modeling critique and sketching a non-uniform pdf. All steps are routine with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{1}{2}(a+b) = 3\) | M1 | Forms one correct equation using \(E(X) = 3\) for rectangular distribution or first principles |
| \(\frac{1}{12}(b-a)^2 = 3\) | M1 | Forms one correct equation using \(\text{Var}(X) = 3\) for rectangular distribution or first principles |
| \(\frac{1}{12}(b-6+b)^2 = 3\) (OE) or \(\frac{1}{12}(6-a-a)^2 = 3\) (OE) | A1 | Forms correct equation in terms of \(a\) or \(b\) only (PI) |
| \(b = 6\), \(a = 0\) | A1 | Correct values of \(a\) and \(b\) |
| \(f(x) = \begin{cases} \frac{1}{6} & 0 \leq x \leq 6 \\ 0 & \text{otherwise} \end{cases}\) | A1F | Fully defined and correct pdf; follow through on their \(a\) and \(b\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The clothes line is more likely to snap in the middle. | E1 | Valid criticism giving reason why clothes line is more or less likely to snap at particular points; accept references to clothes on the line or tension in the line |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Symmetrical pdf with maximum consistent with criticism in (b)(i), with 6 visible | B1F | Do not award for pdfs with non-zero values at \(x >\) their 6 or 6; allow non-symmetrical pdf with clear maximums if criticism in (b)(i) relates to placement of clothes |
# Question 4:
## Part 4(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{1}{2}(a+b) = 3$ | M1 | Forms one correct equation using $E(X) = 3$ for rectangular distribution or first principles |
| $\frac{1}{12}(b-a)^2 = 3$ | M1 | Forms one correct equation using $\text{Var}(X) = 3$ for rectangular distribution or first principles |
| $\frac{1}{12}(b-6+b)^2 = 3$ (OE) or $\frac{1}{12}(6-a-a)^2 = 3$ (OE) | A1 | Forms correct equation in terms of $a$ or $b$ only (PI) |
| $b = 6$, $a = 0$ | A1 | Correct values of $a$ and $b$ |
| $f(x) = \begin{cases} \frac{1}{6} & 0 \leq x \leq 6 \\ 0 & \text{otherwise} \end{cases}$ | A1F | Fully defined and correct pdf; follow through on their $a$ and $b$ |
## Part 4(b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The clothes line is more likely to snap in the middle. | E1 | Valid criticism giving reason why clothes line is more or less likely to snap at particular points; accept references to clothes on the line or tension in the line |
## Part 4(b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Symmetrical pdf with maximum consistent with criticism in (b)(i), with 6 visible | B1F | Do not award for pdfs with non-zero values at $x >$ their 6 or 6; allow non-symmetrical pdf with clear maximums if criticism in (b)(i) relates to placement of clothes |
---4 A random variable $X$ has a rectangular distribution.
The mean of $X$ is 3 and the variance of $X$ is 3\\
4
\begin{enumerate}[label=(\alph*)]
\item Determine the probability density function of $X$.\\
Fully justify your answer.
4
\item A 6 metre clothes line is connected between the point $P$ on one building and the point $Q$ on a second building.
Roy is concerned the clothes line may break. He uses the random variable $X$ to model the distance in metres from $P$ where the clothes line breaks.
4 (b) (i) State a criticism of Roy's model.
4 (b) (ii) On the axes below, sketch the probability density function for an alternative model for the clothes line.\\
\includegraphics[max width=\textwidth, alt={}, center]{3219e2fe-7757-469a-9d0d-654b3e180e8d-05_584_1162_1210_438}
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Statistics 2019 Q4 [7]}}