| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2001 |
| Session | June |
| Marks | 8 |
| 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 z-score calculations and table lookups. Part (a) is routine standardization, while part (b) involves symmetric probability which is a common textbook exercise. Both parts are slightly easier than average A-level questions due to being pure recall/application with no problem-solving insight required. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(Y < 80) = P\left(Z < \frac{\pm(80-100)}{\sqrt{256}}\right)\) | M1 | Standardising; allow \(\sqrt{256}\) or \(256\) |
| \(= P(Z < \pm 1.25)\) | A1 | \(\pm 1.25\) |
| \(= 1 - \Phi(1.25) = \underline{0.1056}\) | A1 (3) | ISW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(100-k \leq Y \leq 100+k) = 0.516\) | ||
| \(\therefore P(Y \leq 100+k) = 0.516 + \frac{1}{2}(1-0.516)\) | ||
| \(= 0.758\) | B1 | \(0.758\) |
| \(\pm \frac{k}{16}\) | B1 | |
| \(\therefore P\left(Z \leq \frac{k}{16}\right) = 0.758\) | M1 | \(\frac{k}{16} = \Phi^{-1}(0.758)\) |
| \(\therefore \frac{k}{16} = 0.7 \Rightarrow \underline{k = 11.2}\) | B1, A1 (5) | \(= 0.7\); \(k = 11.2\) |
## Question 3:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(Y < 80) = P\left(Z < \frac{\pm(80-100)}{\sqrt{256}}\right)$ | M1 | Standardising; allow $\sqrt{256}$ or $256$ |
| $= P(Z < \pm 1.25)$ | A1 | $\pm 1.25$ |
| $= 1 - \Phi(1.25) = \underline{0.1056}$ | A1 (3) | ISW |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(100-k \leq Y \leq 100+k) = 0.516$ | | |
| $\therefore P(Y \leq 100+k) = 0.516 + \frac{1}{2}(1-0.516)$ | | |
| $= 0.758$ | B1 | $0.758$ |
| $\pm \frac{k}{16}$ | B1 | |
| $\therefore P\left(Z \leq \frac{k}{16}\right) = 0.758$ | M1 | $\frac{k}{16} = \Phi^{-1}(0.758)$ |
| $\therefore \frac{k}{16} = 0.7 \Rightarrow \underline{k = 11.2}$ | B1, A1 (5) | $= 0.7$; $k = 11.2$ |
---3. The continuous random variable $Y$ is normally distributed with mean 100 and variance 256 .
\begin{enumerate}[label=(\alph*)]
\item Find $\mathrm { P } ( Y < 80 )$.
\item Find $k$ such that $\mathrm { P } ( 100 - k \leq Y \leq 100 + k ) = 0.516$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2001 Q3 [8]}}