| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Statistics (Further Paper 3 Statistics) |
| Year | 2024 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | One-tail z-test (upper tail) |
| Difficulty | Moderate -0.3 This is a collection of routine Further Maths Statistics questions testing standard procedures: exponential distribution probability (direct formula application), finding median from CDF (solving quadratic equation), Poisson hypothesis test, showing a given mean for a pdf (integration by parts), and a t-test with small sample. All are textbook exercises requiring recall and standard methods with no novel insight, making them slightly easier than average A-level difficulty overall. |
| Spec | 5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration5.03f Relate pdf-cdf: medians and percentiles5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(X) = 7 \times 0.4 + 13 \times 0.35 + 17 \times 0.1 + 21 \times 0.15 = 12.2\) | M1 | Uses correct formula for \(E(X)\); AO 1.1a |
| \(P(X > E(X)) = P(X \geq 13) = 0.35 + 0.1 + 0.15 = 0.6\) | M1 | Obtains either \(P(X > E(X))\) or \(P(X > E(X)) \cap X \leq 17\) for their \(E(X)\); AO 1.1a |
| \(P(X > E(X)) = 0.6\) and \(P(X > E(X)) \cap X \leq 17) = 0.45\); condone incorrect \(E(X)\) leading to these values | A1 | AO 1.1b |
| Uses conditional probability formula \(\dfrac{P(X > E(X)) \cap X \leq 17)}{P(X > E(X))}\) | M1 | AO 1.1a |
| \(P(X \leq 17 \mid X > E(X)) = \dfrac{P(13 \leq X \leq 17)}{P(X \geq 13)} = \dfrac{0.45}{0.6} = 0.75\) | R1 | CSO; must see \(E(X)=12.2\), \(P(X>E(X))=0.6\), and \(P(X>E(X)) \cap X \leq 17)=0.45\); AO 2.1 |
## Question 8:
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(X) = 7 \times 0.4 + 13 \times 0.35 + 17 \times 0.1 + 21 \times 0.15 = 12.2$ | M1 | Uses correct formula for $E(X)$; AO 1.1a |
| $P(X > E(X)) = P(X \geq 13) = 0.35 + 0.1 + 0.15 = 0.6$ | M1 | Obtains either $P(X > E(X))$ or $P(X > E(X)) \cap X \leq 17$ for their $E(X)$; AO 1.1a |
| $P(X > E(X)) = 0.6$ and $P(X > E(X)) \cap X \leq 17) = 0.45$; condone incorrect $E(X)$ leading to these values | A1 | AO 1.1b |
| Uses conditional probability formula $\dfrac{P(X > E(X)) \cap X \leq 17)}{P(X > E(X))}$ | M1 | AO 1.1a |
| $P(X \leq 17 \mid X > E(X)) = \dfrac{P(13 \leq X \leq 17)}{P(X \geq 13)} = \dfrac{0.45}{0.6} = 0.75$ | R1 | CSO; must see $E(X)=12.2$, $P(X>E(X))=0.6$, and $P(X>E(X)) \cap X \leq 17)=0.45$; AO 2.1 |
**Subtotal: 5 marks**
---8\\
16\\
256
2 The random variable $T$ has an exponential distribution with mean 2
Find $\mathrm { P } ( T \leq 1.4 )$\\
Circle your answer.\\
$\mathrm { e } ^ { - 2.8 }$\\
$\mathrm { e } ^ { - 0.7 }$\\
$1 - e ^ { - 0.7 }$\\
$1 - \mathrm { e } ^ { - 2.8 }$ The continuous random variable $Y$ has cumulative distribution function
$$\mathrm { F } ( y ) = \left\{ \begin{array} { l r }
0 & y < 2 \\
- \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\
1 & y \geq 5
\end{array} \right.$$
Find the median of $Y$
Circle your answer.
2\\
$\frac { 10 - 3 \sqrt { 2 } } { 2 }$\\
$\frac { 7 } { 2 }$\\
$\frac { 10 + 3 \sqrt { 2 } } { 2 }$
Turn over for the next question
4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20
Sandra records 162 volcanic eruptions during a period of one week.
Sandra claims that there has been an increase in the mean number of volcanic eruptions per week.
Test Sandra's claim at the $5 \%$ level of significance.\\
5 The continuous random variable $X$ has probability density function
$$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$
Show that the mean of $X$ is $\frac { 3 } { 2 } ( \ln 27 - 2 )$\\
6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old.
Imran claims that the mean retirement age is no longer 29.5 years old.\\
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, $x$. The results are
$$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$
6
\begin{enumerate}[label=(\alph*)]
\item State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim.
6
\item Investigate Imran's claim, using the 10\% level of significance.
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Statistics 2024 Q8 [5]}}