| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Uniform Distribution |
| Type | Name the distribution |
| Difficulty | Easy -1.2 This is a straightforward recall question testing basic knowledge of distributions. Part (a) requires naming the normal distribution and a simple practical reasoning about overfilling. Part (b) requires naming the discrete uniform distribution and applying standard formulas for mean and variance—all routine S1 content with no problem-solving or novel insight required. |
| Spec | 2.04a Discrete probability distributions2.04e Normal distribution: as model N(mu, sigma^2)5.02e Discrete uniform distribution |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) Normal | A1 | |
| (a)(ii) e.g. producer must ensure that most bottles contain at least 75 cl | B1 | |
| (b)(i) Discrete uniform | A1 | |
| (b)(ii) | ||
| \(x\) | 1 | 2 |
| \(P(X=x)\) | \(\frac{1}{4}\) | \(\frac{1}{4}\) |
| Mean \(= \frac{5}{2}\) (symmetry) | A1 | |
| \(E(X^2) = \sum x^2 P(x) = 1 \cdot \frac{1}{4} + 4 \cdot \frac{1}{4} + 9 \cdot \frac{1}{4} + 16 \cdot \frac{1}{4} = \frac{15}{2}\) | M1 A1 | |
| \(\text{Var}(X) = \frac{15}{2} - \left(\frac{5}{2}\right)^2 = \frac{5}{4}\) | M1 A1 | (8 marks total) |
**(a)(i)** Normal | A1 |
**(a)(ii)** e.g. producer must ensure that most bottles contain at least 75 cl | B1 |
**(b)(i)** Discrete uniform | A1 |
**(b)(ii)** | | |
| $x$ | 1 | 2 | 3 | 4 |
| $P(X=x)$ | $\frac{1}{4}$ | $\frac{1}{4}$ | $\frac{1}{4}$ | $\frac{1}{4}$ |
Mean $= \frac{5}{2}$ (symmetry) | A1 |
$E(X^2) = \sum x^2 P(x) = 1 \cdot \frac{1}{4} + 4 \cdot \frac{1}{4} + 9 \cdot \frac{1}{4} + 16 \cdot \frac{1}{4} = \frac{15}{2}$ | M1 A1 |
$\text{Var}(X) = \frac{15}{2} - \left(\frac{5}{2}\right)^2 = \frac{5}{4}$ | M1 A1 | **(8 marks total)**
---\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Name a suitable distribution for modelling the volume of liquid in bottles of wine sold as containing 75 cl.
\item Explain why the mean in such a model would probably be greater than 75 cl.
\end{enumerate}
[2 marks]
\item \begin{enumerate}[label=(\roman*)]
\item Name a suitable distribution for modelling the score on a single throw of a fair four-sided die with the numbers 1, 2, 3 and 4 on its faces.
\item Use your suggested model to find the mean and variance of the score on a single throw of the die.
\end{enumerate}
[6 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q1 [8]}}