| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Expected frequency with unknown parameter |
| Difficulty | Moderate -0.3 This is a straightforward application of normal distribution tables with standard procedures: part (i) is trivial recall, part (ii) requires reverse lookup of z-tables and simple algebra to find μ, and part (iii) applies the found parameter to calculate an expected frequency. All steps are routine with no problem-solving insight required, making it slightly easier than average. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| B1 | 1 mark | Any sensible set of data, must be qualified |
| Answer | Marks | Guidance |
|---|---|---|
| B1, M1 | \(z = \pm 0.64\) seen. Equation relating \(10, \sqrt{21}, 21, \mu\) and their \(z\) or \(1 - \) their \(z\), recognisable \(z\) value ie not \(0.77\) | |
| A1 | 3 marks | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | Standardising, with or without sq rt, no cc, must be their mean | |
| M1 ft | Correct area ie \(< 0.5\), ft on their mean \(> 22\) | |
| M1 | Mult by 300 | |
| A1 | 4 marks | Correct answer, accept 7 or 8 must be integer |
**(i)** Heights, weights, times etc of something
| B1 | 1 mark | Any sensible set of data, must be qualified |
**(ii)** $z = 0.64 = \frac{\mu - 10}{\sqrt{21}}$
$\mu = 12.9$
| B1, M1 | $z = \pm 0.64$ seen. Equation relating $10, \sqrt{21}, 21, \mu$ and their $z$ or $1 - $ their $z$, recognisable $z$ value ie not $0.77$ |
| A1 | 3 marks | Correct answer |
**(iii)** $z = \frac{22 - 12.9}{\sqrt{21}} = 1.986$
$P(X > 22) = 1 - \Phi(1.986) = 1 - 0.9765 = 0.0235$
$300 \times 0.0235 = 7.05$
Answer $= 7$
| M1 | Standardising, with or without sq rt, no cc, must be their mean |
| M1 ft | Correct area ie $< 0.5$, ft on their mean $> 22$ |
| M1 | Mult by 300 |
| A1 | 4 marks | Correct answer, accept 7 or 8 must be integer |
---5 (i) Give an example of a variable in real life which could be modelled by a normal distribution.\\
(ii) The random variable $X$ is normally distributed with mean $\mu$ and variance 21.0. Given that $\mathrm { P } ( X > 10.0 ) = 0.7389$, find the value of $\mu$.\\
(iii) If 300 observations are taken at random from the distribution in part (ii), estimate how many of these would be greater than 22.0.
\hfill \mbox{\textit{CAIE S1 2006 Q5 [8]}}