| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2004 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Interval probability P(a < X < b) |
| Difficulty | Easy -1.2 Part (a) is pure recall of basic properties of the normal distribution. Part (b) is a straightforward standardization and table lookup for a symmetric interval around the mean, requiring only routine application of the z-score formula. This is a standard introductory S1 question with no problem-solving element. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Symmetrical (about the mean \(\mu\)) | B1 | Any 3 sensible properties |
| Mode = mean = median | B1 | |
| Horizontal axis asymptotic to curve | B1 | Accept: bell shaped sketch, 95% of data lies within 2 s.d.s of the mean, 68% between \(\mu \pm \sigma\), most of data within 3 s.d. of mean, never touches axes at either side |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(X \sim N(27, 10^2)\); \(P(26 < x < 28) = P\left(\frac{26-27}{10} < Z < \frac{28-27}{10}\right)\) | M1 | Standardising with \(\mu = 27\), \(\sigma = 10\) or \(\sqrt{10}\) |
| One correct z-value seen | A1 | |
| \(= P(-0.1 < Z < 0.1)\) | A1 | \(-0.1\) or \(0.1\) |
| \(= \Phi(0.1) - \{1 - \Phi(0.1)\}\) or \(2 \times \{\Phi(0.1) - 0.5\}\) | ||
| \(= \underline{0.0796}\) | A1 | \(0.0796\) or \(0.0797\) |
## Question 2:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Symmetrical (about the mean $\mu$) | B1 | Any 3 sensible properties |
| Mode = mean = median | B1 | |
| Horizontal axis asymptotic to curve | B1 | Accept: bell shaped sketch, 95% of data lies within 2 s.d.s of the mean, 68% between $\mu \pm \sigma$, most of data within 3 s.d. of mean, never touches axes at either side | **(3)** |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $X \sim N(27, 10^2)$; $P(26 < x < 28) = P\left(\frac{26-27}{10} < Z < \frac{28-27}{10}\right)$ | M1 | Standardising with $\mu = 27$, $\sigma = 10$ or $\sqrt{10}$ |
| One correct z-value seen | A1 | |
| $= P(-0.1 < Z < 0.1)$ | A1 | $-0.1$ or $0.1$ |
| $= \Phi(0.1) - \{1 - \Phi(0.1)\}$ or $2 \times \{\Phi(0.1) - 0.5\}$ | | |
| $= \underline{0.0796}$ | A1 | $0.0796$ or $0.0797$ | **(4)** |
---2. The random variable $X$ is normally distributed with mean $\mu$ and variance $\sigma ^ { 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Write down 3 properties of the distribution of $X$.
Given that $\mu = 27$ and $\sigma = 10$
\item find $\mathrm { P } ( 26 < X < 28 )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2004 Q2 [7]}}