| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Finding unknown boundaries |
| Difficulty | Standard +0.3 This is a straightforward normal distribution problem requiring standardization and inverse normal calculations. Part (i) is routine (finding P(X > 70) using z-scores), while part (ii) requires understanding that short buildings comprise 1/3 of the non-tall buildings and using inverse normal tables—slightly above average due to the two-step reasoning in part (ii), but still a standard S1 question type. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(P(\text{tall}) = P\left(z > \frac{70-50}{16}\right) = P(z > 1.25)\) | M1 | \(+ve/-ve\) Standardising no cc no sq rt no sq |
| \(= 1 - 0.8944 = 0.106\) | A1 [2] | Correct answer |
| (ii) \(P(\text{short}) = (1 - 0.1056)/3\) | M1 | Subt their (i) from 1 or their (i) and multiplying by \(\frac{1}{3}\) or \(\frac{2}{3}\) |
| \(= 0.2981\) | A1 ft | Rounding to 0.298, only ft for \(\frac{(1-\text{(i)})}{3}\) |
| \(z = -0.53\) | A1 | \(\pm\) z-value rounding to 0.53, condone \(\pm 0.24\) |
| \(-0.53 = \frac{x-50}{16}\) | M1 | Standardising with their z value (not a probability), no cc sq rt etc. |
| \(x = 41.5\) | A1 [5] | Correct answer |
**(i)** $P(\text{tall}) = P\left(z > \frac{70-50}{16}\right) = P(z > 1.25)$ | M1 | $+ve/-ve$ Standardising no cc no sq rt no sq
$= 1 - 0.8944 = 0.106$ | A1 [2] | Correct answer
**(ii)** $P(\text{short}) = (1 - 0.1056)/3$ | M1 | Subt their (i) from 1 or their (i) and multiplying by $\frac{1}{3}$ or $\frac{2}{3}$
$= 0.2981$ | A1 ft | Rounding to 0.298, only ft for $\frac{(1-\text{(i)})}{3}$
$z = -0.53$ | A1 | $\pm$ z-value rounding to 0.53, condone $\pm 0.24$
$-0.53 = \frac{x-50}{16}$ | M1 | Standardising with their z value (not a probability), no cc sq rt etc.
$x = 41.5$ | A1 [5] | Correct answer3 Buildings in a certain city centre are classified by height as tall, medium or short. The heights can be modelled by a normal distribution with mean 50 metres and standard deviation 16 metres. Buildings with a height of more than 70 metres are classified as tall.\\
(i) Find the probability that a building chosen at random is classified as tall.\\
(ii) The rest of the buildings are classified as medium and short in such a way that there are twice as many medium buildings as there are short ones. Find the height below which buildings are classified as short.
\hfill \mbox{\textit{CAIE S1 2013 Q3 [7]}}