| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Probability calculation plus find unknown boundary |
| Difficulty | Moderate -0.3 This is a straightforward S1 normal distribution question requiring standard techniques: converting to z-scores using tables for parts (a) and (b), and working backwards from a probability to find the mean in part (c). All three parts are routine applications of the normal distribution with no conceptual challenges, making it slightly easier than average for A-level. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(Z < \frac{38.2 - 32.5}{\sqrt{18.6}}) = P(Z < 1.32) = 0.9066\) | M2 A1 | |
| \(P(\frac{31 - 32.5}{\sqrt{18.6}} < Z < \frac{35 - 32.5}{\sqrt{18.6}}) = P(-0.35 < Z < 0.58)\) | M2 | |
| \(= P(Z < 0.58) - P(Z < 0.35) = 0.7190 - 0.3632 = 0.3558\) | M1 A1 | |
| \(P(Z > \frac{110 - \mu}{7.2}) = 0.138\) | M1 | |
| \(\frac{110 - \mu}{7.2} = 1.09\); \(\mu = 102\) (3sf) | M1 A2 | (11) |
| $P(Z < \frac{38.2 - 32.5}{\sqrt{18.6}}) = P(Z < 1.32) = 0.9066$ | M2 A1 | |
| $P(\frac{31 - 32.5}{\sqrt{18.6}} < Z < \frac{35 - 32.5}{\sqrt{18.6}}) = P(-0.35 < Z < 0.58)$ | M2 | |
| $= P(Z < 0.58) - P(Z < 0.35) = 0.7190 - 0.3632 = 0.3558$ | M1 A1 | |
| $P(Z > \frac{110 - \mu}{7.2}) = 0.138$ | M1 | |
| $\frac{110 - \mu}{7.2} = 1.09$; $\mu = 102$ (3sf) | M1 A2 | (11) |4. The random variable $A$ is normally distributed with a mean of 32.5 and a variance of 18.6
Find
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { P } ( A < 38.2 )$,
\item $\mathrm { P } ( 31 \leq A \leq 35 )$,
The random variable $B$ is normally distributed with a standard deviation of 7.2\\
Given also that $\mathrm { P } ( B > 110 ) = 0.138$,
\item find the mean of $B$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q4 [11]}}