| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2013 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Comparison involving sums or multiples |
| Difficulty | Standard +0.8 This S3 question requires understanding of linear combinations of independent normal variables and forming correct compound distributions (L vs 3S, and L vs 3S with different coefficients). Part (b) particularly requires algebraic manipulation to express '3 times' as a single normal variable. While the calculations are standard once set up, the conceptual step of forming L - 3S as a new normal variable and correctly handling variances (not standard deviations) elevates this above routine questions. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
\begin{enumerate}
\item Blumen is a perfume sold in bottles. The amount of perfume in each bottle is normally distributed. The amount of perfume in a large bottle has mean 50 ml and standard deviation 5 ml . The amount of perfume in a small bottle has mean 15 ml and standard deviation 3 ml .
\end{enumerate}
One large and 3 small bottles of Blumen are chosen at random.\\
(a) Find the probability that the amount in the large bottle is less than the total amount in the 3 small bottles.
A large bottle and a small bottle of Blumen are chosen at random.\\
(b) Find the probability that the large bottle contains more than 3 times the amount in the small bottle.\\
\hfill \mbox{\textit{Edexcel S3 2013 Q5 [12]}}