| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Major (Further Statistics Major) |
| Year | 2021 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find multiple parameters from system |
| Difficulty | Challenging +1.8 This is a challenging Further Maths statistics question requiring: (1) using continuity and E(X)=2 to solve a system for two parameters involving piecewise integration, (2) finding the median by solving F(x)=0.5 (likely requiring numerical/algebraic work across pieces), and (3) applying CLT for sample mean distribution. The multi-step parameter finding and piecewise integration across two domains with different forms elevates this significantly above standard A-level, though it follows established techniques without requiring novel insight. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| 11 | (a) | F(3)=1⇒∫ 2 ax2 dx+∫ 3 b(3−x)2 dx=1 |
| Answer | Marks |
|---|---|
| 8 | M1 |
| Answer | Marks |
|---|---|
| [5] | 3.1a |
| Answer | Marks | Guidance |
|---|---|---|
| 11 | (b) | F(2)=∫ 2 1x2 dx= 1 |
| Answer | Marks |
|---|---|
| 4 | B1 |
| Answer | Marks |
|---|---|
| [3] | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | Or m=3−3 3 |
| Answer | Marks | Guidance |
|---|---|---|
| 11 | (c) | ( 0.2) |
| Answer | Marks |
|---|---|
| Estimate P(Mean < 1.9) = 0.0569 | M1 |
| Answer | Marks |
|---|---|
| [3] | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | For use of Normal distribution |
Question 11:
11 | (a) | F(3)=1⇒∫ 2 ax2 dx+∫ 3 b(3−x)2 dx=1
0 2
⇒ 8a+1b=1
3 3
E(X)=2⇒∫ 2 ax3dx+∫ 3 bx(3−x)2 dx=2
0 2
⇒4a+ 3b=2
4
a= 1, b=2
8 | M1
A1
M1
A1
A1
[5] | 3.1a
1.1
3.1a
1.1
1.1
11 | (b) | F(2)=∫ 2 1x2 dx= 1
0 8 3
⇒∫ m 2(3−x)2 dx= 1
2 6
⇒−2(3−m)3+ 2 = 1
3 3 6
⇒(3−m)3 = 3 ⇒m=2.09 (2.0914…)
4 | B1
M1
A1
[3] | 3.1a
2.2a
1.1 | Or m=3−3 3
4
11 | (c) | ( 0.2)
Using N 2,
50
N(2, 0.004)
Estimate P(Mean < 1.9) = 0.0569 | M1
M1
A1
[3] | 3.1a
1.1a
1.1 | For use of Normal distribution
For correct values
PMT
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For staff training purposes and as part of our quality assurance programme your call may be
recorded or monitored11 The continuous random variable $X$ has probability density function given by\\
$f ( x ) = \begin{cases} a x ^ { 2 } & 0 \leqslant x < 2 , \\ b ( 3 - x ) ^ { 2 } & 2 \leqslant x \leqslant 3 , \\ 0 & \text { otherwise } \end{cases}$\\
where $a$ and $b$ are positive constants.
\begin{enumerate}[label=(\alph*)]
\item Given that $\mathrm { E } ( X ) = 2$, determine the values of $a$ and $b$.
\item Determine the median value of $X$.
\item A random sample of 50 observations of $X$ is selected.
Given that $\operatorname { Var } ( X ) = 0.2$, determine an estimate of the probability that the mean value of the 50 observations is less than 1.9.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Major 2021 Q11 [11]}}