| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Session | Specimen |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Outliers and box plots |
| Difficulty | Moderate -0.3 This S1 question combines routine normal distribution calculations (parts a-c using tables and symmetry) with straightforward application of the outlier formula (parts d-e). While it requires multiple steps and connects different concepts (normal distribution and box plot outliers), each individual step is standard and the question provides all necessary formulas, making it slightly easier than average overall. |
| Spec | 2.02h Recognize outliers2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| END |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(D>20) = P\!\left(Z > \dfrac{20-30}{8}\right)\) | M1 | Attempt to standardise 20 or 40 using 30 and 8 |
| \(= P(Z > -1.25)\) | A1 | \(z = \pm 1.25\) |
| \(= \mathbf{0.8944}\) awrt 0.894 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(D < Q_3) = 0.75\) so \(\dfrac{Q_3 - 30}{8} = 0.67\) | M1 B1 | M1 for setting \(\frac{Q_3-30}{8}\) equal to a \(z\) value; B1 for \(z\) value between 0.67–0.675; M0 for 0.7734 on RHS |
| \(Q_3 = \) awrt \(\mathbf{35.4}\) | A1 | M1B0A1 for use of \(z=0.68\) in correct expression with awrt 35.4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(35.4 - 30 = 5.4\), so \(Q_1 = 30 - 5.4 =\) awrt \(\mathbf{24.6}\) | B1ft | Follow through using their quartile values |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(Q_3 - Q_1 = 10.8\), so \(1.5(Q_3-Q_1)=16.2\), giving \(Q_1 - 16.2 = h\) or \(Q_3 + 16.2 = k\) | M1 | Attempt \(1.5(\text{IQR})\) and add/subtract using formulae; follow through their quartiles |
| \(h = \mathbf{8.4}\) to \(\mathbf{8.6}\) and \(k = \mathbf{51.4}\) to \(\mathbf{51.6}\) | A1 | Both required |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(2P(D > 51.6) = 2P(Z > 2.7)\) | M1 | Attempt \(2P(D > \text{their } k)\) or \(P(D>\text{their }k)+P(D<\text{their }h)\) |
| \(= 2[1 - 0.9965] =\) awrt \(\mathbf{0.007}\) | M1 A1 | 2nd M1 for standardising \(h\) or \(k\); require both M's for A mark |
## Question 7:
**Part (a)**
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(D>20) = P\!\left(Z > \dfrac{20-30}{8}\right)$ | M1 | Attempt to standardise 20 or 40 using 30 and 8 |
| $= P(Z > -1.25)$ | A1 | $z = \pm 1.25$ |
| $= \mathbf{0.8944}$ awrt 0.894 | A1 | |
**Part (b)**
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(D < Q_3) = 0.75$ so $\dfrac{Q_3 - 30}{8} = 0.67$ | M1 B1 | M1 for setting $\frac{Q_3-30}{8}$ equal to a $z$ value; B1 for $z$ value between 0.67–0.675; M0 for 0.7734 on RHS |
| $Q_3 = $ awrt $\mathbf{35.4}$ | A1 | M1B0A1 for use of $z=0.68$ in correct expression with awrt 35.4 |
**Part (c)**
| Answer | Mark | Guidance |
|--------|------|----------|
| $35.4 - 30 = 5.4$, so $Q_1 = 30 - 5.4 =$ awrt $\mathbf{24.6}$ | B1ft | Follow through using their quartile values |
**Part (d)**
| Answer | Mark | Guidance |
|--------|------|----------|
| $Q_3 - Q_1 = 10.8$, so $1.5(Q_3-Q_1)=16.2$, giving $Q_1 - 16.2 = h$ or $Q_3 + 16.2 = k$ | M1 | Attempt $1.5(\text{IQR})$ and add/subtract using formulae; follow through their quartiles |
| $h = \mathbf{8.4}$ to $\mathbf{8.6}$ and $k = \mathbf{51.4}$ to $\mathbf{51.6}$ | A1 | Both required |
**Part (e)**
| Answer | Mark | Guidance |
|--------|------|----------|
| $2P(D > 51.6) = 2P(Z > 2.7)$ | M1 | Attempt $2P(D > \text{their } k)$ or $P(D>\text{their }k)+P(D<\text{their }h)$ |
| $= 2[1 - 0.9965] =$ awrt $\mathbf{0.007}$ | M1 A1 | 2nd M1 for standardising $h$ or $k$; require both M's for A mark |\begin{enumerate}
\item The distances travelled to work, $D \mathrm {~km}$, by the employees at a large company are normally distributed with $D \sim \mathrm {~N} \left( 30,8 ^ { 2 } \right)$.\\
(a) Find the probability that a randomly selected employee has a journey to work of more than 20 km .\\
(b) Find the upper quartile, $Q _ { 3 }$, of $D$.\\
(c) Write down the lower quartile, $Q _ { 1 }$, of $D$.
\end{enumerate}
An outlier is defined as any value of $D$ such that $D < h$ or $D > k$ where
$$h = Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \quad \text { and } \quad k = Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
(d) Find the value of $h$ and the value of $k$.
An employee is selected at random.\\
(e) Find the probability that the distance travelled to work by this employee is an outlier.\\
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\hfill \mbox{\textit{Edexcel S1 Q7 [12]}}