| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Linear relationship μ = kσ |
| Difficulty | Standard +0.3 Part (i) involves standard normal distribution calculations (z-scores, inverse normal, symmetric intervals) that are routine A-level exercises. Part (ii) requires recognizing that the relationship μ = 3σ allows standardization to eliminate the parameter, which is a modest conceptual step beyond routine but still well within typical A-level problem-solving. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.0478\) or \(0.048\) (2 sf) | B1 [1] | AO 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(22.5\) or \(23\) (2 sf) | B1 [1] | AO 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X<20+b)=0.75\) or \(P(X>20+b)=0.25\) | M1 | AO 1.1a |
| \(20+b=22.02...\) or \(22.0\) or \(22\) | A1 | AO 1.1 |
| \(b=2.02\) or \(2.0\) (2 sf) Allow \(b=2\) | A1 [3] | AO 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1.5\mu-\mu}{\mu/3}\) | M1 | AO 1.1a |
| \(=\frac{3}{2}\) | A1 | AO 1.1 |
| \(P(X>1.5\mu)=0.0668\) or \(0.067\) (2 sf) | A1 [3] | AO 1.1 |
## Question 8:
### Part (i)(a):
$0.0478$ or $0.048$ (2 sf) | **B1** [1] | AO 1.1 | BC
### Part (i)(b):
$22.5$ or $23$ (2 sf) | **B1** [1] | AO 1.1 | BC
### Part (i)(c):
$P(X<20+b)=0.75$ or $P(X>20+b)=0.25$ | **M1** | AO 1.1a |
$20+b=22.02...$ or $22.0$ or $22$ | **A1** | AO 1.1 |
$b=2.02$ or $2.0$ (2 sf) Allow $b=2$ | **A1** [3] | AO 1.1 |
T&I method: Try 2 values, one $\approx 2$ M1; Correct probs for two values in [2, 2.1] A1 (0.495 & 0.516); Correct probs for two values in [2, 2.05] & ans 2.0 or 2 A1
### Part (ii):
$\frac{1.5\mu-\mu}{\mu/3}$ | **M1** | AO 1.1a | $\frac{4.5\sigma-3\sigma}{\sigma}$; SC (eg) Let $\mu=1$; $N(1,\frac{1}{9})$ M1
$=\frac{3}{2}$ | **A1** | AO 1.1 | $X=\frac{3}{2}$ A0
$P(X>1.5\mu)=0.0668$ or $0.067$ (2 sf) | **A1** [3] | AO 1.1 | $P(X>\frac{3}{2})=0.067$ A1
---8 (i) The variable $X$ has the distribution $\mathrm { N } ( 20,9 )$.
\begin{enumerate}[label=(\alph*)]
\item Find $\mathrm { P } ( X > 25 )$.
\item Given that $\mathrm { P } ( X > a ) = 0.2$, find $a$.
\item Find $b$ such that $\mathrm { P } ( 20 - b < X < 20 + b ) = 0.5$.\\
(ii) The variable $Y$ has the distribution $\mathrm { N } \left( \mu , \frac { \mu ^ { 2 } } { 9 } \right)$. Find $\mathrm { P } ( Y > 1.5 \mu )$.
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2018 Q8 [8]}}