| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2007 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Direct comparison with scalar multiple (different variables) |
| Difficulty | Standard +0.8 This question requires understanding of linear combinations of normal variables and constructing new random variables (L - 4S, sum of 4S values, |L-T|). Part (a) requires forming L - 4S and recognizing it's normally distributed, part (b) is straightforward application of sum properties, but part (c) requires converting an absolute value inequality into a probability statement P(-0.1 < L-T < 0.1). While the techniques are standard for S3, the multi-step reasoning and need to construct appropriate combinations makes this moderately above average difficulty. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
7. A set of scaffolding poles come in two sizes, long and short. The length $L$ of a long pole has the normal distribution $\mathrm { N } \left( 19.7,0.5 ^ { 2 } \right)$. The length $S$ of a short pole has the normal distribution $\mathrm { N } \left( 4.9,0.2 ^ { 2 } \right)$. The random variables $L$ and $S$ are independent.
A long pole and a short pole are selected at random.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that the length of the long pole is more than 4 times the length of the short pole.
Four short poles are selected at random and placed end to end in a row. The random variable $T$ represents the length of the row.
\item Find the distribution of $T$.
\item Find $\mathrm { P } ( | L - T | < 0.1 )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2007 Q7 [15]}}