| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Quadratic relationship μ = kσ² |
| Difficulty | Challenging +1.2 Part (a) requires solving simultaneous equations involving a quadratic relationship and using inverse normal tables with standardization, which is non-routine for S1. Part (b) is a standard symmetric probability calculation. The combination of algebraic manipulation with normal distribution and the quadratic constraint elevates this above typical textbook exercises. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(z > \frac{2\mu - \mu}{\sigma} = \frac{\mu}{3\sigma} = \frac{7\sigma}{3} = 1.272\) | M1 | Standardising attempt resulting in \(z > \text{some } \mu/\sigma\) |
| M1 | Substituting to eliminate \(\mu\) or \(\sigma\) | |
| B1 | 1.272 seen | |
| A1 [4] | Both answers correct |
| Answer | Marks |
|---|---|
| \(\mu = 0.693\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(z = 0.674\) | M1 | Using 0.75 o.e |
| A1 | \(\pm 0.674\) seen | |
| \(\frac{a + 33 - 33}{\sqrt{21}} = 0.674\) | M1 | Standardising, no cc, must have sq rt |
| \(a = 3.09\) | A1 [4] | Correct answer |
**(a)** $z > \frac{2\mu - \mu}{\sigma} = \frac{\mu}{3\sigma} = \frac{7\sigma}{3} = 1.272$ | M1 | Standardising attempt resulting in $z > \text{some } \mu/\sigma$
| M1 | Substituting to eliminate $\mu$ or $\sigma$
| B1 | 1.272 seen
| A1 [4] | Both answers correct
$\sigma = 0.545$
$\mu = 0.693$ | |
**(b)** $P(X < a + 33) = 0.75$
$z = 0.674$ | M1 | Using 0.75 o.e
| A1 | $\pm 0.674$ seen
$\frac{a + 33 - 33}{\sqrt{21}} = 0.674$ | M1 | Standardising, no cc, must have sq rt
$a = 3.09$ | A1 [4] | Correct answer5
\begin{enumerate}[label=(\alph*)]
\item The random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$. It is given that $3 \mu = 7 \sigma ^ { 2 }$ and that $\mathrm { P } ( X > 2 \mu ) = 0.1016$. Find $\mu$ and $\sigma$.
\item It is given that $Y \sim \mathrm {~N} ( 33,21 )$. Find the value of $a$ given that $\mathrm { P } ( 33 - a < Y < 33 + a ) = 0.5$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2011 Q5 [8]}}