| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2006 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Comparison involving sums or multiples |
| Difficulty | Standard +0.3 This is a straightforward application of linear combinations of normal distributions with clear setup and standard calculations. Part (a) requires routine formulas for sum of independent normals (mean = 7×78.5 + 8×62.0, variance = 7×12.6² + 8×9.8²), part (b) tests understanding of independence assumption, and part (c) is a standard normal probability calculation. While it involves multiple steps, each is algorithmic with no problem-solving insight required, making it slightly easier than average. |
| Spec | 5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(W) = 7 \times 78.5 + 6 \times 62.0 = 1045.50\) | M1A1 | awrt (OSO M1A1) |
| \(Var(W) = 7 \times 12.6^1 + 8 \times 9.8^2 = 1879.64\) | M1A1 | (BFO M1A1) |
| Answer | Marks | Guidance |
|---|---|---|
| Independent | B1 | (used in Variance formula) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(W > 1090) = P\left(Z > \frac{1090-1045.50}{\sqrt{1879.64}}\right)\) | M1 | |
| \(= P(Z > 1.03)\) | awrt 1.07 A1 | |
| \(= 1 - 0.8485\) | 1 - M1 | |
| \(= 0.1515\) | A1 | (4) |
| AWRT (0.152) | (9) |
## Part (a)
$W = M_1 + \cdots + M_7 + F_1 + \cdots + F_8$
$E(W) = 7 \times 78.5 + 6 \times 62.0 = 1045.50$ | M1A1 | awrt (OSO M1A1)
$Var(W) = 7 \times 12.6^1 + 8 \times 9.8^2 = 1879.64$ | M1A1 | (BFO M1A1)
## Part (b)
Independent | B1 | (used in Variance formula)
## Part (c)
$P(W > 1090) = P\left(Z > \frac{1090-1045.50}{\sqrt{1879.64}}\right)$ | M1 |
$= P(Z > 1.03)$ | awrt 1.07 A1 |
$= 1 - 0.8485$ | 1 - M1 |
$= 0.1515$ | A1 | (4)
| | AWRT (0.152) | | (9)
---The workers in a large office block use a lift that can carry a maximum load of 1090 kg. The weights of the male workers are normally distributed with mean 78.5 kg and standard deviation 12.6 kg. The weights of the female workers are normally distributed with mean 62.0 kg and standard deviation 9.8 kg.
Random samples of 7 males and 8 females can enter the lift.
\begin{enumerate}[label=(\alph*)]
\item Find the mean and variance of the total weight of the 15 people that enter the lift. [4]
\item Comment on any relationship you have assumed in part (a) between the two samples. [1]
\item Find the probability that the maximum load of the lift will be exceeded by the total weight of the 15 people. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2006 Q5 [9]}}