| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Estimate from percentile/frequency data |
| Difficulty | Challenging +1.2 This requires setting up and solving simultaneous equations from two percentiles (11% and 40% from z-tables), which is more demanding than single-percentile problems. However, it's a standard S1 technique with straightforward algebra once the z-values are identified, and part (b) is routine application. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(P(X < 30) = 0.11\) where \(\frac{30 - \mu}{\sigma} = -1.23\) so \(30 - \mu = -1.23\sigma\) | M1 A1 A1 | |
| \(P(X > 90) = 0.4\) where \(\frac{90 - \mu}{\sigma} = 0.25\) so \(90 - \mu = 0.25\sigma\) | M1 A1 A1 | |
| \(1.48\sigma = 60\) giving \(\sigma = 40.5\), \(\mu = 79.9\) | M1 A1 A1 | |
| (b) \(P(X > 100) = P(Z > (100 - 79.9)/40.5) = P(Z > 0.50) = 1 - 0.692 = 0.308\), so would expect 308 | M1 A1 M1 A1 | 13 marks |
(a) $P(X < 30) = 0.11$ where $\frac{30 - \mu}{\sigma} = -1.23$ so $30 - \mu = -1.23\sigma$ | M1 A1 A1 |
$P(X > 90) = 0.4$ where $\frac{90 - \mu}{\sigma} = 0.25$ so $90 - \mu = 0.25\sigma$ | M1 A1 A1 |
$1.48\sigma = 60$ giving $\sigma = 40.5$, $\mu = 79.9$ | M1 A1 A1 |
(b) $P(X > 100) = P(Z > (100 - 79.9)/40.5) = P(Z > 0.50) = 1 - 0.692 = 0.308$, so would expect 308 | M1 A1 M1 A1 | 13 marks4. A botanist believes that the lengths of the branches on trees of a certain species can be modelled by a normal distribution.\\
When he measures the lengths of 500 branches, he finds 55 which are less than 30 cm long and 200 which are more than 90 cm long.
\begin{enumerate}[label=(\alph*)]
\item Find the mean and the standard deviation of the lengths.
\item In a sample of 1000 branches, how many would he expect to find with lengths greater than 1 metre?
\section*{STATISTICS 1 (A) TEST PAPER 7 Page 2}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q4 [13]}}