| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2018 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Symmetric properties of normal |
| Difficulty | Moderate -0.8 This question tests basic understanding of normal distribution symmetry and the 68-95-99.7 rule. Part (a) requires recognizing that P(μ < Y < 17) = 0.5 - 0.4 = 0.1 by symmetry, while part (b) uses the standard result that P(μ - σ < Y < μ) = 0.34. Both parts are direct applications of fundamental properties with minimal calculation, 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \([P(\mu < Y < 17) =] \ 0.5 - 0.4 = \mathbf{0.1}\) | B1 | For 0.1 as clearly their final answer or clear statement "\(P(\mu < Y < 17) = 0.1\)". Ignore poor or incorrect notation if answers are correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(Y > \mu - \sigma) = P(Z > -1)\) | M1 | For an attempt to standardise \(\mu - \sigma\), allow for \(\pm\frac{(\mu-\sigma)-\mu}{\sigma}\), can be un-simplified |
| \(= 0.841(3)\) | A1 | For 0.841 or better (calc 0.84134473...) or \(1 - 0.8413... = 0.1587\) (accept 0.159). Sight of 0.841(3) or 0.1587 or 0.159 (or better) scores M1 A1. May be statement e.g. \(P(Y > \mu - \sigma) = 0.841(3)\) or on clearly labelled diagram |
| \(P(\mu - \sigma < Y < 17) = 0.8413 - 0.4\) | dM1 | Dep on 1st M1, for a correct use of their 0.8413 and the given 0.4. Or \(0.341(3) +\) their (a). Or \(0.6 -\) their 0.1587 |
| \(= \mathbf{0.441}(3)\) | A1 | For 0.441 or better (correct answer only 4/4) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(Y > \mu - \sigma) = P(Z > -1)\) | M1 | Standardise \(\mu - \sigma\) (and may get \(z = -1\)) scores 1st M1 as in scheme |
| \(P(Y>17) = 0.4 \Rightarrow Z = \left[\frac{17-\mu}{\sigma}\right] = 0.25(33471...)\) so need \(P(-1 < Z < 0.25)\) | dM1 | Use inv' normal to get \(\frac{17-\mu}{\sigma} = 0.25(33471...)\) and write/attempt \(P(-1 < Z < 0.25..)\) |
| Sight of \(P(-1 < Z < 0.253...)\) | 1st A1 | Write or attempt \(P(-1 < Z < 0.253...)\) also scores 1st A1 (need 0.253 or better). NB Just standardising and getting 0.2533 etc is no use unless it is part of a correct probability statement that would lead to the final answer |
| \(= \mathbf{0.441}(3)\) | 2nd A1 |
# Question 3:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $[P(\mu < Y < 17) =] \ 0.5 - 0.4 = \mathbf{0.1}$ | B1 | For 0.1 as clearly their final answer or clear statement "$P(\mu < Y < 17) = 0.1$". Ignore poor or incorrect notation if answers are correct |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(Y > \mu - \sigma) = P(Z > -1)$ | M1 | For an attempt to standardise $\mu - \sigma$, allow for $\pm\frac{(\mu-\sigma)-\mu}{\sigma}$, can be un-simplified |
| $= 0.841(3)$ | A1 | For 0.841 or better (calc 0.84134473...) or $1 - 0.8413... = 0.1587$ (accept 0.159). Sight of 0.841(3) or 0.1587 or 0.159 (or better) scores M1 A1. May be statement e.g. $P(Y > \mu - \sigma) = 0.841(3)$ or on clearly labelled diagram |
| $P(\mu - \sigma < Y < 17) = 0.8413 - 0.4$ | dM1 | Dep on 1st M1, for a correct use of their 0.8413 and the given 0.4. Or $0.341(3) +$ their (a). Or $0.6 -$ their 0.1587 |
| $= \mathbf{0.441}(3)$ | A1 | For 0.441 or better (correct answer only 4/4) |
## ALT Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(Y > \mu - \sigma) = P(Z > -1)$ | M1 | Standardise $\mu - \sigma$ (and may get $z = -1$) scores 1st M1 as in scheme |
| $P(Y>17) = 0.4 \Rightarrow Z = \left[\frac{17-\mu}{\sigma}\right] = 0.25(33471...)$ so need $P(-1 < Z < 0.25)$ | dM1 | Use inv' normal to get $\frac{17-\mu}{\sigma} = 0.25(33471...)$ and write/attempt $P(-1 < Z < 0.25..)$ |
| Sight of $P(-1 < Z < 0.253...)$ | 1st A1 | Write or attempt $P(-1 < Z < 0.253...)$ also scores 1st A1 (need 0.253 or better). NB Just standardising and getting 0.2533 etc is no use unless it is part of a correct probability statement that would lead to the final answer |
| $= \mathbf{0.441}(3)$ | 2nd A1 | |
---3. The random variable $Y$ has a normal distribution with mean $\mu$ and standard deviation $\sigma$ The $\mathrm { P } ( Y > 17 ) = 0.4$
Find
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { P } ( \mu < Y < 17 )$
\item $\mathrm { P } ( \mu - \sigma < Y < 17 )$
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2018 Q3 [5]}}