| Exam Board | WJEC |
|---|---|
| Module | Further Unit 5 (Further Unit 5) |
| Year | 2024 |
| Session | June |
| Marks | 19 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Distribution of sample mean |
| Difficulty | Standard +0.8 This is a multi-part Further Maths statistics question requiring inverse normal calculations, sampling distributions, comparison of sums of normal variables, and reverse-engineering a parameter from a given probability. While each individual technique is standard (parts a-c are routine), part (d) requires working backwards from P(X > 3E) = 0.35208 to find σ, which involves setting up X - 3E ~ N(μ, σ²) correctly and solving, plus critically evaluating the reasonableness of the result. The combination of multiple normal distribution concepts and the non-routine reverse calculation elevates this above average difficulty. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | 19 | |
| Total | 80 | |
| Group A | 32 | 8 |
| Group B | 30 | 29 |
| Answer | Marks |
|---|---|
| number | Additional page, if required. |
| Answer | Marks |
|---|---|
| number | Additional page, if required. |
Question 7:
7 | 19
Total | 80
Group A | 32 | 8 | 24 | 16 | 10 | 20 | 22 | 18 | 23 | 21 | 26 | 14
Group B | 30 | 29 | 11 | 25 | 38 | 36 | 28 | 12 | 17
Question
number | Additional page, if required.
Write the question number(s) in the left-hand margin.
Question
number | Additional page, if required.
Write the question number(s) in the left-hand margin.A farmer uses many identical containers to store four different types of grain: wheat, corn, einkorn and emmer.
\begin{enumerate}[label=(\alph*)]
\item The mass $W$, in kg, of wheat stored in each individual container is normally distributed with mean $\mu$ and standard deviation 0.6. Given that, for containers of wheat, 10\% store less than 19 kg, find the value of $\mu$. [3]
\end{enumerate}
The mass $X$, in kg, of corn stored in each individual container is normally distributed with mean 20.1 and standard deviation 1.2.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the probability that the mean mass of corn in a random sample of 8 containers of corn will be greater than 20 kg. [3]
\end{enumerate}
The mass $Y$, in kg, of einkorn stored in each individual container is normally distributed with mean 22.2 and standard deviation 1.5.
The farmer and his wife need to move two identical wheelbarrows, one of which is loaded with 3 containers of corn, and the other of which is loaded with 3 containers of einkorn. They agree that the farmer's wife will move the heavier wheelbarrow.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Calculate the probability that the farmer's wife will move
\begin{enumerate}[label=(\roman*)]
\item the einkorn,
\item the corn. [5]
\end{enumerate}
\item The mass $E$, in kg, of emmer stored in each individual container is normally distributed with mean 10.5 and standard deviation $\sigma$. The farmer's son tries to calculate the probability that the mass of corn in a single container will be more than three times the mass of emmer in a single container. He obtains an answer of 0.35208.
\begin{enumerate}[label=(\roman*)]
\item Find the value of $\sigma$ that the farmer's son used.
\item Explain why the value of $\sigma$ that he used is unreasonable. [8]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 5 2024 Q7 [19]}}