| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Linear relationship μ = kσ |
| Difficulty | Standard +0.3 This question tests standard normal distribution calculations with algebraic manipulation. Part (a) requires standardizing X and using tables (routine). Part (b) involves reverse lookup from probability to z-score and then another standardization. While it requires careful algebraic manipulation and understanding of the normal distribution, these are standard S1 techniques with no novel problem-solving required—slightly above average due to the algebraic component but still straightforward for well-prepared students. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(P(X < 2\mu) = P\left[Z < \frac{2\mu - \mu}{2\mu/3}\right] = P(Z < 1.5) = 0.933\) | M1 A1 A1 M1 A1 | |
| (b) (i) \(P(Z < 2\mu/\sigma) = 0.86\), \(2\mu/\sigma = 1.08\), \(\mu = 0.54\sigma\) | M1 A1 M1 A1 | |
| (ii) \(P(X > 0) = P\left[Z > -\frac{\mu}{0.54}\right] = P(Z > -0.54) = 0.705\) | M1 A1 M1 A1 | Total: 13 |
(a) $P(X < 2\mu) = P\left[Z < \frac{2\mu - \mu}{2\mu/3}\right] = P(Z < 1.5) = 0.933$ | M1 A1 A1 M1 A1 |
(b) (i) $P(Z < 2\mu/\sigma) = 0.86$, $2\mu/\sigma = 1.08$, $\mu = 0.54\sigma$ | M1 A1 M1 A1 |
(ii) $P(X > 0) = P\left[Z > -\frac{\mu}{0.54}\right] = P(Z > -0.54) = 0.705$ | M1 A1 M1 A1 | **Total: 13**The random variable $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$.
\begin{enumerate}[label=(\alph*)]
\item If $2\mu = 3\sigma$, find P$(X < 2\mu)$. [5 marks]
\item If, instead, P$(X < 3\mu) = 0.86$,
\begin{enumerate}[label=(\roman*)]
\item find $\mu$ in terms of $\sigma$, [4 marks]
\item calculate P$(X > 0)$. [4 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q4 [13]}}