Moderate -0.3 These are standard Further Maths Statistics questions requiring direct application of formulas: exponential distribution CDF (recognizing λ=1/2 and using standard formula), finding median by solving F(y)=0.5 (straightforward quadratic), and a routine Poisson hypothesis test. All are textbook exercises with no novel insight required, though the Further Maths context places them slightly above average C1/C2 difficulty.
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8
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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.
States both hypotheses using correct language; may use \(\lambda\) or \(\mu\)
Test statistic calculation
Answer
Marks
Guidance
\(P(X \geq 162) = \mathbf{AWRT}\ 0.037\) or \(P(X \leq 161) = \mathbf{AWRT}\ 0.963\) or \(P(X > 162) = \mathbf{AWRT}\ 0.031\) or \(P(X \leq 162) = \mathbf{AWRT}\ 0.969\) or \(P(X \geq 161) = \mathbf{AWRT}\ 0.044\) or correct critical region
M1
Uses Poisson model \(X \sim \text{Po}(140)\)
\(P(X \geq 162) = \mathbf{AWRT}\ 0.037\) or correct critical region \(X \geq 161\)
A1
Comparison
Answer
Marks
Guidance
\(0.037 < 0.05\), Reject \(H_0\)
M1
Correctly comparing probability with 0.05 (or 0.025 if two-tailed), or comparing 162 with critical region
Conclusion (inference)
Answer
Marks
Guidance
Reject \(H_0\)
A1F
FT comparison; condone Accept \(H_1\)
Contextual conclusion
Answer
Marks
Guidance
Sufficient evidence to suggest that the mean number of volcanic eruptions per week has increased
R1
Must not be definite; e.g. use of 'suggest', 'support'; must refer to mean number of volcanic eruptions per week
## Question 4:
**Hypotheses**
$H_0: \lambda = 140$, $H_1: \lambda > 140$, $X \sim \text{Po}(140)$ | B1 | States both hypotheses using correct language; may use $\lambda$ or $\mu$
**Test statistic calculation**
$P(X \geq 162) = \mathbf{AWRT}\ 0.037$ or $P(X \leq 161) = \mathbf{AWRT}\ 0.963$ or $P(X > 162) = \mathbf{AWRT}\ 0.031$ or $P(X \leq 162) = \mathbf{AWRT}\ 0.969$ or $P(X \geq 161) = \mathbf{AWRT}\ 0.044$ or correct critical region | M1 | Uses Poisson model $X \sim \text{Po}(140)$
$P(X \geq 162) = \mathbf{AWRT}\ 0.037$ or correct critical region $X \geq 161$ | A1 |
**Comparison**
$0.037 < 0.05$, Reject $H_0$ | M1 | Correctly comparing probability with 0.05 (or 0.025 if two-tailed), or comparing 162 with critical region
**Conclusion (inference)**
Reject $H_0$ | A1F | FT comparison; condone Accept $H_1$
**Contextual conclusion**
Sufficient evidence to suggest that the mean number of volcanic eruptions per week has increased | R1 | Must not be definite; e.g. use of 'suggest', 'support'; must refer to mean number of volcanic eruptions per week
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4\\
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.\\
\hfill \mbox{\textit{AQA Further Paper 3 Statistics 2024 Q4 [6]}}