| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2015 |
| Session | June |
| Marks | 4 |
| Topic | Normal Distribution |
| Type | Mixed calculations with boundaries |
| Difficulty | Moderate -0.5 This is a straightforward application of normal distribution with standard table lookups and inverse normal calculations. Part (i) requires a single z-score calculation and table lookup, while part (ii) involves finding a percentile—both are routine procedures covered in any stats course. The context is clear and no conceptual insight is needed beyond applying the standard normal distribution formulas. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
**Question 3**
**(i)** $T \sim \text{N}(43.2,\ 6.3^2)$; Require $P(T < 50)$
$= P\!\left(Z < \frac{50 - 43.2}{6.3} = 1.079(3...)\right)$
$= 0.8598$
— M1: Formulate the problem. M1: Standardising. A1: c.a.o. $Z$ value. A1: From tables. Ft *their* $Z$ value. Must involve use of difference columns. **[4]**
**(ii)** $\frac{T - 43.2}{6.3} = 1.645$
— M1: Set up equation for $T$. B1: 1.645 seen.
$\therefore T = 43.2 + 1.645 \times 6.3 = 53.56$
$60 - 53.56 = 6.44\ \text{(min)}$
$\therefore$ Jack should leave by 08 06
— A1: c.a.o. A1: Interpret as time of day. Accept 08 07. **[4]**3 Jack's journey time, in minutes, to work each morning is modelled by the normal distribution $\mathrm { N } \left( 43.2,6.3 ^ { 2 } \right)$.\\
(i) If Jack leaves home at 0810 , find the probability that he arrives at work by 0900 .\\
(ii) Find the time by which Jack should leave home in order to be at least $95 \%$ certain that he arrives at work by 0900 .
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2015 Q3 [4]}}