| Exam Board | SPS |
|---|---|
| Module | SPS FM Statistics (SPS FM Statistics) |
| Year | 2026 |
| Session | January |
| Marks | 8 |
| Topic | Linear combinations of normal random variables |
| Type | Multiple stage process probability |
| Difficulty | Standard +0.3 This is a straightforward application of normal distribution properties and linear combinations. Part (a) is basic standardization, part (b) requires knowing that sums of independent normals are normal (with variance addition), and part (c) involves finding the distribution of a difference. All techniques are standard for Further Maths Statistics with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Colour | Mean | Standard deviation |
| Red | 20 | 0.8 |
| Yellow | 30 | 0.9 |
| Green | 50 | 1.2 |
2.
At a toy factory, wooden blocks of approximate heights $20 \mathrm {~mm} , 30 \mathrm {~mm}$ and 50 mm are made in red, yellow and green respectively. The heights of the blocks in mm are modelled by independent random variables which are Normally distributed with means and standard deviations as shown in the table.
\begin{center}
\begin{tabular}{ | l | c | c | }
\hline
Colour & Mean & Standard deviation \\
\hline
Red & 20 & 0.8 \\
\hline
Yellow & 30 & 0.9 \\
\hline
Green & 50 & 1.2 \\
\hline
\end{tabular}
\end{center}
In parts (a), (b) and (c), the blocks are selected randomly and independently of one another.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that the height of a red block is less than 19 mm .
\item A tower is made of 15 blocks stacked on top of each other consisting of 5 red blocks, 5 yellow blocks and 5 green blocks.
Determine the probability that the tower is at least 495 mm high.
\item Determine the probability that a tower made of 3 red blocks will be at least 1 mm higher than a tower made of 2 yellow blocks.\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Statistics 2026 Q2 [8]}}