| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Linear relationship μ = kσ |
| Difficulty | Standard +0.3 This is a straightforward normal distribution problem requiring standardization to z-scores and using tables. Part (i) involves simple algebraic substitution and one table lookup. Part (ii) requires working backwards from a probability to find a z-value, then solving a linear equation. Both parts are routine applications of standard techniques with no conceptual challenges beyond basic S1 material. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(P(X < 2\mu) = P\left(z < \frac{2\mu - \mu}{\sigma}\right) = P(z < 5/3) = 0.952\) | M1, A1, A1 [3] | Standardising, and attempt to get 1 variable, no cc, no \(\sqrt{}\), no sq. \(\pm 5/3\) seen oe. Rounding to correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mu = -1.57\sigma\) | M1, B1, A1 [3] | standardising attempt resulting in \(z = -\) some \(\mu/\sigma\). allow \(\pm\left(\frac{\mu/3 - \mu}{\sigma}\right)\). \(\pm 1.047\) seen. correct single number, answer must have a minus sign and \(\mu = \ldots.\sigma\) |
(i) $P(X < 2\mu) = P\left(z < \frac{2\mu - \mu}{\sigma}\right) = P(z < 5/3) = 0.952$ | M1, A1, A1 [3] | Standardising, and attempt to get 1 variable, no cc, no $\sqrt{}$, no sq. $\pm 5/3$ seen oe. Rounding to correct answer
(ii) $P\left(X < \frac{\mu}{3}\right) = P\left(z < \frac{-2\mu}{3\sigma}\right)$
$\frac{-2\mu}{3\sigma} = -1.047$
$\mu = -1.57\sigma$ | M1, B1, A1 [3] | standardising attempt resulting in $z = -$ some $\mu/\sigma$. allow $\pm\left(\frac{\mu/3 - \mu}{\sigma}\right)$. $\pm 1.047$ seen. correct single number, answer must have a minus sign and $\mu = \ldots.\sigma$
---The random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$.
\begin{enumerate}[label=(\roman*)]
\item Given that $5\sigma = 3\mu$, find $\mathrm{P}(X < 2\mu)$. [3]
\item With a different relationship between $\mu$ and $\sigma$, it is given that $\mathrm{P}(X < \frac{4\mu}{3}) = 0.8524$. Express $\mu$ in terms of $\sigma$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2010 Q4 [6]}}