| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2016 |
| Session | June |
| Marks | 4 |
| Topic | Linear combinations of normal random variables |
| Type | Comparison involving sums or multiples |
| Difficulty | Standard +0.8 Part (i) requires converting a cost constraint to a mass threshold and standardizing a normal variable (routine). Part (ii) involves forming linear combinations of independent normals (4B vs 3S), finding the distribution of their difference, and computing a probability—this requires solid understanding of variance rules for independent variables and is more demanding than typical A-level statistics questions, though still a standard Further Maths technique. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.04b Linear combinations: of normal distributions |
(i) $\text{N}(120, \ldots)$
$\sigma^2 = 0.8^2 \times 1200\ [= 768]$
$1 - \Phi\!\left(\dfrac{140-120}{\sqrt{768}}\right) = \mathbf{0.235}(3)$
- M1: Normal, mean 120 or 1.20
- M1: Allow $0.8 \times 1200$ etc
- A1: Both parameters correct
- A1: Answer, in range $[0.235, 0.236]$
OR: $P(\geq 175)$ from $\text{N}(150, 200)$: M1; $\dfrac{175-150}{\sqrt{1200}}$: A1; $0.235(3)$: A2
**[4]**
(ii) $B_1 + B_2 + B_3 + B_4 - S_1 - S_2 - S_3 \sim \text{N}(60, \ldots)$
Variance $4\times1200 + 3\times1500 = 9300$
$\Phi\!\left(\dfrac{0-60}{\sqrt{9300}}\right) = \Phi(-0.622) = \mathbf{0.267}$
- M1: Consider $\pm(B_1+B_2+B_3+B_4-S_1-S_2-S_3)$ or $4B-3S$
- M1: Normal, mean 60
- A1: Correct variance
- A1: Answer, a.r.t. 0.267 [0.2699] [NB: $\Phi(-60/\sqrt{33700}) = 0.3700$ is 2/4]
**[4]**2 The mass in grams of a pre-cut piece of Brie cheese is a random variable with the distribution $\mathrm { N } ( 150,1200 )$. Brie costs 80 p per 100 g .\\
(i) Find the probability that a randomly chosen piece of Brie costs more than $\pounds 1.40$.
The mass in grams of a pre-cut piece of Stilton cheese is an independent random variable with the distribution $\mathrm { N } ( 180,1500 )$.\\
(ii) Find the probability that the total mass of four randomly chosen pieces of Brie is less than the total mass of three randomly chosen pieces of Stilton.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2016 Q2 [4]}}