| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Outliers and box plots |
| Difficulty | Standard +0.3 This is a straightforward S1 question requiring standard normal distribution calculations (probability, quartiles) followed by applying given formulas for outliers. All steps are routine: standardize for part (a), use inverse normal for parts (b-c), substitute into given formulas for part (d), and calculate tail probabilities for part (e). No novel insight or complex problem-solving required, just methodical application of techniques. |
| Spec | 2.02h Recognize outliers2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
**Question 7(a):** M1 for an attempt to standardise 20 or 40 using 30 and 8. 1st A1 for $z = \pm 1.25$. 2nd A1 for awrt 0.894.
**Question 7(b):** M1 for $\frac{Q_3 - 30}{8} = $ to a $z$ value. M0 for 0.7734 on RHS. B1 for ($z$ value) between 0.67–0.675 seen. M1B0A1 for use of $z = 0.68$ in correct expression with awrt 35.4.
**Question 7(e):** 1st M1 for attempting 2P(D > their k) or (P(D > their k)+P(D < their h)). 2nd M1 for standardising their $h$ or $k$ (may have missed the 2) so allow for standardising P(D > 8.4) or P(D < 51.6). Require booths Ms to award an A mark.7. The distances travelled to work, $D \mathrm {~km}$, by the employees at a large company are normally distributed with $D \sim \mathrm {~N} \left( 30,8 ^ { 2 } \right)$.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that a randomly selected employee has a journey to work of more than 20 km .
\item Find the upper quartile, $Q _ { 3 }$, of $D$.
\item Write down the lower quartile, $Q _ { 1 }$, of $D$.
An outlier is defined as any value of $D$ such that $D < h$ or $D > k$ where
$$h = Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \quad \text { and } \quad k = Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
\item Find the value of $h$ and the value of $k$.
An employee is selected at random.
\item Find the probability that the distance travelled to work by this employee is an outlier.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2010 Q7 [12]}}