| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Quadratic relationship μ = kσ² |
| Difficulty | Challenging +1.2 Part (a) is a routine symmetric probability calculation requiring inverse normal tables. Part (b) is more challenging, requiring students to set up two equations (the quadratic relationship and a z-score equation from the probability) and solve simultaneously, but the algebraic manipulation is straightforward once the setup is recognized. This is above average due to the non-standard constraint linking μ and σ, but well within reach for competent S1 students. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{\pm q}{7.4}\) or \(\frac{2q}{7.4} = z\) or *probability* (o.e) | M1 | Rounding to ± 0.58 or ± 0.15 seen |
| M1 | Standardising, no cc, no sq, no sq rt | |
| \(q = 4.31\) | A1 3 | correct answer |
| (b) \(\frac{0.5\mu - \mu}{\sigma} = \frac{\pm 0.5\mu}{\sigma}\) | M1 | Standardising attempt some \(\mu/\sigma\) allow cc, sq rt, sq. Can be implied |
| \(\frac{0.2\sigma^2}{\sigma} = -0.2\sigma = -0.580\) | B1 | ± 0.580 seen (accept ±0.58) |
| M1 | substituting to eliminate \(\mu\) or \(\sigma\), arriving at numerical solution, any \(z\) value or probability – not dependent |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mu = 3.36\) | A1 4 | both answers correct, accept 2.9 |
(a) $P(X < a + 82) = 0.72$
$z = 0.583$
$\frac{\pm q}{7.4}$ or $\frac{2q}{7.4} = z$ or *probability* (o.e) | M1 | Rounding to ± 0.58 or ± 0.15 seen
| M1 | Standardising, no cc, no sq, no sq rt
$q = 4.31$ | A1 3 | correct answer
(b) $\frac{0.5\mu - \mu}{\sigma} = \frac{\pm 0.5\mu}{\sigma}$ | M1 | Standardising attempt some $\mu/\sigma$ allow cc, sq rt, sq. Can be implied
$\frac{0.2\sigma^2}{\sigma} = -0.2\sigma = -0.580$ | B1 | ± 0.580 seen (accept ±0.58)
| M1 | substituting to eliminate $\mu$ or $\sigma$, arriving at numerical solution, any $z$ value or probability – not dependent
$\sigma = 2.90$
$\mu = 3.36$ | A1 4 | both answers correct, accept 2.95
\begin{enumerate}[label=(\alph*)]
\item The random variable $X$ is normally distributed with mean 82 and standard deviation 7.4. Find the value of $q$ such that $\mathrm { P } ( 82 - q < X < 82 + q ) = 0.44$.
\item The random variable $Y$ is normally distributed with mean $\mu$ and standard deviation $\sigma$. It is given that $5 \mu = 2 \sigma ^ { 2 }$ and that $\mathrm { P } \left( Y < \frac { 1 } { 2 } \mu \right) = 0.281$. Find the values of $\mu$ and $\sigma$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2013 Q5 [7]}}