| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2024 |
| Session | January |
| Marks | 14 |
| 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 normal variables with multiple operations (sum, difference, scalar multiplication) across three parts. Part (a) is standard but part (b) requires forming 1.8X - Y > 0 which is less routine, and part (c) involves comparing one container to the mean of three, requiring careful variance calculation of X₁ - (X₁+X₂+X₃)/3. The multi-step reasoning and non-standard setups elevate this above typical S3 exercises. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
\begin{enumerate}
\item Small containers and large containers are independently filled with fruit juice.
\end{enumerate}
The amounts of fruit juice in small containers are normally distributed with mean 180 ml and standard deviation 4.5 ml
The amounts of fruit juice in large containers are normally distributed with mean 330 ml and standard deviation 6.7 ml
The random variable $W$ represents the total amount of fruit juice in a random sample of 2 small containers minus the amount of fruit juice in 1 randomly selected large container.\\
$W \sim \mathrm {~N} ( a , b )$ where $a$ and $b$ are positive constants.\\
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$ and the value of $b$
\item Find the probability that a randomly chosen large container of fruit juice contains more than 1.8 times the amount of fruit juice in a randomly chosen small container.
A random sample of 3 small containers of fruit juice is taken.
\item Find the probability that the first container of fruit juice in this sample contains at least 5 ml more than the mean amount of fruit juice in all 3 small containers.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2024 Q7 [14]}}