| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2002 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Mixed calculations with boundaries |
| Difficulty | Moderate -0.8 This is a straightforward application of normal distribution with standard bookwork: (i) requires finding P(30 < X < 40) using tables/calculator, and (ii) involves finding an inverse normal value for the 90th percentile. Both parts are routine S1 techniques with no conceptual challenges or multi-step reasoning required. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| Part (i): \(z = \pm \frac{40 - 35.0}{11.6} = \pm 0.431\) | M1 | For standardising (\(\sqrt{11.6}\) in denom M1, cc M0 11.6² M0) |
| M1 | For subtracting two relevant probabilities or equivalent | |
| A1 | 3 | For correct answer |
| \(\Phi(0.431) - \{1 - \Phi(0.431)\} = 0.334\) | ||
| Part (ii): \(z = \pm 1.282\) or \(\pm 1.281\) only | B1 | For stating \(z\) |
| \(1.282 = \frac{x - 35.0}{11.6}\) | M1 | For solving an equation for \(x\) with some \(z\) value from tables, allow cc, \(\sqrt{11.6}\), 35-x, not \(11.6^2\) |
| \(x = 49.9\) or \(49.8\) on \(z = 1.28\) | A1 | 3 |
**Part (i):** $z = \pm \frac{40 - 35.0}{11.6} = \pm 0.431$ | M1 | For standardising ($\sqrt{11.6}$ in denom M1, cc M0 11.6² M0)
| M1 | For subtracting two relevant probabilities or equivalent
| A1 | 3 | For correct answer
$\Phi(0.431) - \{1 - \Phi(0.431)\} = 0.334$ | |
**Part (ii):** $z = \pm 1.282$ or $\pm 1.281$ only | B1 | For stating $z$
$1.282 = \frac{x - 35.0}{11.6}$ | M1 | For solving an equation for $x$ with some $z$ value from tables, allow cc, $\sqrt{11.6}$, 35-x, not $11.6^2$
$x = 49.9$ or $49.8$ on $z = 1.28$ | A1 | 3 | For correct answer
---3 The distance in metres that a ball can be thrown by pupils at a particular school follows a normal distribution with mean 35.0 m and standard deviation 11.6 m .\\
(i) Find the probability that a randomly chosen pupil can throw a ball between 30 and 40 m .\\
(ii) The school gives a certificate to the $10 \%$ of pupils who throw further than a certain distance. Find the least distance that must be thrown to qualify for a certificate.
\hfill \mbox{\textit{CAIE S1 2002 Q3 [6]}}