| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Direct comparison of probabilities |
| Difficulty | Moderate -0.3 This is a straightforward application of normal distribution with standardization to find probabilities and use of inverse normal tables. Part (i) requires two routine z-score calculations, and part (ii) is a standard reverse lookup problem. The question involves no conceptual challenges beyond basic normal distribution mechanics, making it slightly easier than average for A-level statistics. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{Zotoc: } z = \frac{367-320}{21.6} = 2.176\) | M1 | Standardising either car's fuel, no cc, no sq, no √ |
| \(\text{Gannor: } z = \frac{367-350}{7.5} = 2.267\) | ||
| \(P(\text{Zotoc}) = 0.985\) | A1 | Correct answer |
| \(P(\text{Gannor}) = 0.988\) | A1 [3] | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(z = 0.23\) | B1 | ± 0.23 seen |
| \(0.23 = \frac{x-320}{21.6}\) | M1 | Standardising either car, no cc, no sq rt, no sq |
| \(x = 324.968\) | M1nd | \(320 + d - 320\) i.e. just \(d\) on num |
| \(d = 4.97\) | A1 [4] | Correct answer, –4.97 gets A0 |
**(i)**
$\text{Zotoc: } z = \frac{367-320}{21.6} = 2.176$ | M1 | Standardising either car's fuel, no cc, no sq, no √
$\text{Gannor: } z = \frac{367-350}{7.5} = 2.267$ | |
$P(\text{Zotoc}) = 0.985$ | A1 | Correct answer
$P(\text{Gannor}) = 0.988$ | A1 [3] | Correct answer
**(ii)**
$z = 0.23$ | B1 | ± 0.23 seen
$0.23 = \frac{x-320}{21.6}$ | M1 | Standardising either car, no cc, no sq rt, no sq
$x = 324.968$ | M1nd | $320 + d - 320$ i.e. just $d$ on num
$d = 4.97$ | A1 [4] | Correct answer, –4.97 gets A05 The distance the Zotoc car can travel on 20 litres of fuel is normally distributed with mean 320 km and standard deviation 21.6 km . The distance the Ganmor car can travel on 20 litres of fuel is normally distributed with mean 350 km and standard deviation 7.5 km . Both cars are filled with 20 litres of fuel and are driven towards a place 367 km away.\\
(i) For each car, find the probability that it runs out of fuel before it has travelled 367 km .\\
(ii) The probability that a Zotoc car can travel at least $( 320 + d ) \mathrm { km }$ on 20 litres of fuel is 0.409 . Find the value of $d$.
\hfill \mbox{\textit{CAIE S1 2010 Q5 [7]}}