| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Probability calculation plus find unknown boundary |
| Difficulty | Standard +0.3 This is a straightforward application of normal distribution with inverse normal calculations. Part (a) requires finding σ using P(X > 1.9) = 0.02 and standardizing, while part (b) uses the found σ to determine a new width. Both parts follow standard S1 procedures with no conceptual challenges beyond routine inverse normal table usage. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks |
|---|---|
| (a) \(P(Z > 1.9) = 0.02\) | \(M1\) \(A1\) |
| \(P(Z > 0.3/\sigma) = 0.02\) | \(M1\) \(A1\) |
| \(\sigma = 0.3/2.06 = 0.146\) | |
| (b) If \(P(X < x) = 0.995\), \((x - 1.6)/0.146 = 2.60\) | \(M1\) \(A1\) \(M1\) \(A1\) |
| \(x = 1.98\) m |
**(a)** $P(Z > 1.9) = 0.02$ | $M1$ $A1$ |
$P(Z > 0.3/\sigma) = 0.02$ | $M1$ $A1$ |
$\sigma = 0.3/2.06 = 0.146$ | |
**(b)** If $P(X < x) = 0.995$, $(x - 1.6)/0.146 = 2.60$ | $M1$ $A1$ $M1$ $A1$ |
$x = 1.98$ m | |
**Total: 8 marks**The entrance to a car park is $1.9$ m wide. It is found that this is too narrow for $2\%$ of the vehicles which need to use the car park. The widths of these vehicles are modelled by a normal distribution with mean $1.6$ m.
\begin{enumerate}[label=(\alph*)]
\item Find the standard deviation of the distribution. [4 marks]
\end{enumerate}
It is decided to widen the entrance so that $99.5\%$ of vehicles will be able to use it.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the minimum width needed to achieve this. [4 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q3 [8]}}