| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2009 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Probability calculation plus find unknown boundary |
| Difficulty | Moderate -0.8 This is a straightforward S1 normal distribution question requiring standard z-score calculations and table lookups. Part (a) is direct standardization, parts (b) and (c) are inverse normal problems using tables, and part (d) follows immediately from symmetry. All techniques are routine for this specification with no problem-solving or conceptual 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 | Marks | Guidance |
| \(P(X < 39) = P\left(Z < \frac{39-30}{5}\right)\) | M1 | For standardising with \(\sigma\), \(z = \pm\frac{39-30}{5}\) is OK |
| \(= P(Z < 1.8) = \underline{0.9641}\) | A1 | For 0.9641 or awrt 0.964 but if they go on to calculate \(1 - 0.9641\) they get M1A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(X < d) = P\left(Z < \frac{d-30}{5}\right) = 0.1151\) | ||
| \(1 - 0.1151 = 0.8849\) | M1 | For attempting \(1 - 0.1151\). Must be seen in (b) in connection with finding \(d\) |
| (allow \(\pm 1.2\)) | B1 | For \(z = \pm 1.2\). They must state \(z = \pm 1.2\) or imply it is a \(z\) value by its use. This mark is only available in part (b). |
| \(\Rightarrow z = -1.2\) | M1A1 | \(2^{\text{nd}}\) M1 for \(\left(\frac{d-30}{5}\right) =\) their negative \(z\) value (or equivalent) |
| \(\therefore \frac{d-30}{5} = -1.2 \quad \underline{d = 24}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(X > e) = 0.1151\) so \(e = \mu + (\mu - \text{their } d)\) or \(\frac{e-30}{5} = 1.2\) or \(-\text{their } z\) | M1 | For a full method to find \(e\). If they used \(z = 1.2\) in (b) they can get M1 for \(z = \pm 1.2\) here. If they use symmetry about the mean \(\mu + (\mu - \text{their } d)\) then ft their \(d\) for M1. Must explicitly see the method used unless the answer is correct. |
| \(\underline{e = 36}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(d < X < e) = 1 - 2 \times 0.1151\) | M1 | For a complete method or use of a correct expression e.g. "their \(0.8849\)" \(- 0.1151\), or if their \(d <\) their \(e\) using their values with \(P(X \leq e) - P(X \leq d)\). If their \(d \geq\) their \(e\) then they can only score from an argument like \(1 - 2\times0.1151\). A negative probability or probability \(> 1\) for part (d) scores M0A0 |
| \(= 0.7698\) AWRT \(\underline{0.770}\) | A1 |
# Question 6:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(X < 39) = P\left(Z < \frac{39-30}{5}\right)$ | M1 | For standardising with $\sigma$, $z = \pm\frac{39-30}{5}$ is OK |
| $= P(Z < 1.8) = \underline{0.9641}$ | A1 | For 0.9641 or awrt 0.964 but if they go on to calculate $1 - 0.9641$ they get M1A0 |
**(2 marks)**
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(X < d) = P\left(Z < \frac{d-30}{5}\right) = 0.1151$ | | |
| $1 - 0.1151 = 0.8849$ | M1 | For attempting $1 - 0.1151$. Must be seen in (b) in connection with finding $d$ |
| (allow $\pm 1.2$) | B1 | For $z = \pm 1.2$. They must state $z = \pm 1.2$ or imply it is a $z$ value by its use. This mark is only available in part (b). |
| $\Rightarrow z = -1.2$ | M1A1 | $2^{\text{nd}}$ M1 for $\left(\frac{d-30}{5}\right) =$ their negative $z$ value (or equivalent) |
| $\therefore \frac{d-30}{5} = -1.2 \quad \underline{d = 24}$ | | |
**(4 marks)**
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(X > e) = 0.1151$ so $e = \mu + (\mu - \text{their } d)$ or $\frac{e-30}{5} = 1.2$ or $-\text{their } z$ | M1 | For a full method to find $e$. If they used $z = 1.2$ in (b) they can get M1 for $z = \pm 1.2$ here. If they use symmetry about the mean $\mu + (\mu - \text{their } d)$ then ft their $d$ for M1. Must explicitly see the method used unless the answer is correct. |
| $\underline{e = 36}$ | A1 | |
**(2 marks)**
## Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(d < X < e) = 1 - 2 \times 0.1151$ | M1 | For a complete method or use of a correct expression e.g. "their $0.8849$" $- 0.1151$, or if their $d <$ their $e$ using their values with $P(X \leq e) - P(X \leq d)$. If their $d \geq$ their $e$ then they can only score from an argument like $1 - 2\times0.1151$. A negative probability or probability $> 1$ for part (d) scores M0A0 |
| $= 0.7698$ AWRT $\underline{0.770}$ | A1 | |
**(2 marks)**
**[Total: 10 marks]**6. The random variable $X$ has a normal distribution with mean 30 and standard deviation 5 .
\begin{enumerate}[label=(\alph*)]
\item Find $\mathrm { P } ( X < 39 )$.
\item Find the value of $d$ such that $\mathrm { P } ( X < d ) = 0.1151$
\item Find the value of $e$ such that $\mathrm { P } ( X > e ) = 0.1151$
\item Find $\mathrm { P } ( d < X < e )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2009 Q6 [10]}}