| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | October |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Symmetric properties of normal |
| Difficulty | Moderate -0.8 This question tests understanding of normal distribution symmetry through straightforward application of the property that the normal distribution is symmetric about its mean. Part (a) requires recognizing symmetry, part (b) uses complement rules, and part (c) applies conditional probability with the results from (a) and (b). All parts are direct applications of basic properties with no problem-solving insight required, making this 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 |
|---|---|---|
| 1(a) | \(P(X > \mu - a) = 0.65\) | B1 |
| 1(b) | \(P(\mu - a < X < \mu + a) = 1 - 2 \times 0.35\) or \("0.65" - 0.35\) or \(0.15 + 0.15 = 0.3\) | M1, A1 |
| 1(c) | \(P(X < \mu + a | X > \mu - a) = \frac{"(b)"}{" (a)"} = \frac{0.3}{0.65} = \frac{6}{13}\) (Allow awrt 0.462) |
**1(a)** | $P(X > \mu - a) = 0.65$ | B1 | For 0.65
**1(b)** | $P(\mu - a < X < \mu + a) = 1 - 2 \times 0.35$ or $"0.65" - 0.35$ or $0.15 + 0.15 = 0.3$ | M1, A1 | M1 for correct numerical expression; A1 for 0.3 (Answer only scores both marks)
**1(c)** | $P(X < \mu + a | X > \mu - a) = \frac{"(b)"}{" (a)"} = \frac{0.3}{0.65} = \frac{6}{13}$ (Allow awrt 0.462) | M1, A1 | M1 for correct ratio of probabilities or follow through their answers provided (b) < (a); A1 for $\frac{6}{13}$ or an exact equivalent and allow awrt 0.462
---\begin{enumerate}
\item The random variable $X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$
\end{enumerate}
Given that $\mathrm { P } ( X > \mu + a ) = 0.35$ where $a$ is a constant, find\\
(a) $\mathrm { P } ( X > \mu - a )$\\
(b) $\mathrm { P } ( \mu - a < X < \mu + a )$\\
(c) $\mathrm { P } ( X < \mu + a \mid X > \mu - a )$\\
\hfill \mbox{\textit{Edexcel S1 2016 Q1 [5]}}