| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Standard two probabilities given |
| Difficulty | Standard +0.3 This is a standard S2 question requiring inverse normal calculations to find μ and σ from two given probabilities, followed by a straightforward interpretation about distribution shape. The mechanics are routine (standardize, use z-tables, solve simultaneous equations), though slightly above average difficulty due to the algebraic manipulation required and the conceptual part (ii) about skewness. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{140 - \mu}{\sigma} = -2.326\); \(\frac{300 - \mu}{\sigma} = 0.842\) | M1, B1, A1 | One standardisation equated to \(\Phi^{-1}\), allow "1−", \(\sigma^2\). Both 2.33 and 0.84 at least, ignore signs |
| Solve to obtain: \(\mu = 257.49\); \(\sigma = 50.51\) | M1, A1, A1 | Both equations completely correct, \(\checkmark\) on their \(z\). Both 2.33 and 0.84 at least, ignore signs. Solve two simultaneous equations to find one variable. \(\mu\) value, in range [257, 258]. \(\sigma\) in range [50.4, 50.55] |
| (ii) Higher; as there is positive skew | B1, B1 | "Higher" or equivalent stated. Plausible reason, allow from normal calculations |
(i) $\frac{140 - \mu}{\sigma} = -2.326$; $\frac{300 - \mu}{\sigma} = 0.842$ | M1, B1, A1 | One standardisation equated to $\Phi^{-1}$, allow "1−", $\sigma^2$. Both 2.33 and 0.84 at least, ignore signs
Solve to obtain: $\mu = 257.49$; $\sigma = 50.51$ | M1, A1, A1 | Both equations completely correct, $\checkmark$ on their $z$. Both 2.33 and 0.84 at least, ignore signs. Solve two simultaneous equations to find one variable. $\mu$ value, in range [257, 258]. $\sigma$ in range [50.4, 50.55]
(ii) Higher; as there is positive skew | B1, B1 | "Higher" or equivalent stated. Plausible reason, allow from normal calculations
---3 The continuous random variable $T$ has mean $\mu$ and standard deviation $\sigma$. It is known that $\mathrm { P } ( T < 140 ) = 0.01$ and $\mathrm { P } ( T < 300 ) = 0.8$.\\
(i) Assuming that $T$ is normally distributed, calculate the values of $\mu$ and $\sigma$.
In fact, $T$ represents the time, in minutes, taken by a randomly chosen runner in a public marathon, in which about $10 \%$ of runners took longer than 400 minutes.\\
(ii) State with a reason whether the mean of $T$ would be higher than, equal to, or lower than the value calculated in part (i).
\hfill \mbox{\textit{OCR S2 2006 Q3 [8]}}