| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Mixed calculations with boundaries |
| Difficulty | Moderate -0.3 This is a standard normal distribution question requiring routine z-score calculations and inverse normal lookups. Part (a) is straightforward probability calculation, part (b) uses symmetry and percentage points, and part (c) involves working backwards from a probability to find standard deviation. All techniques are core S1 material with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.05d Confidence intervals: using normal distribution |
| Answer | Marks |
|---|---|
| (a) \(P(5 < X < 9) = P(-1.5 < Z < -0.5) = 0.6915 - 0.0668 = 0.625\) | M1 A1 M1 A1 |
| (b) Need \(P(X < k) = 0.9\), so \((k-8)/2 = 1.28\), so \(k = 10.56\) | B1 M1 A1 |
| Range is 5·4 hours to 10·6 hours | A1 A1 |
| (c) If \(P(X > 5) = 0.8\), then \((5-8)/\sigma = -0.84\), so \(\sigma = 3.57\) | M1 A1 A1 |
| 12 marks total |
(a) $P(5 < X < 9) = P(-1.5 < Z < -0.5) = 0.6915 - 0.0668 = 0.625$ | M1 A1 M1 A1 |
(b) Need $P(X < k) = 0.9$, so $(k-8)/2 = 1.28$, so $k = 10.56$ | B1 M1 A1 |
Range is 5·4 hours to 10·6 hours | A1 A1 |
(c) If $P(X > 5) = 0.8$, then $(5-8)/\sigma = -0.84$, so $\sigma = 3.57$ | M1 A1 A1 |
| 12 marks total |The times taken by a group of people to complete a task are modelled by a normal distribution with mean 8 hours and standard deviation 2 hours.
Use this model to calculate
\begin{enumerate}[label=(\alph*)]
\item the probability that a person chosen at random took between 5 and 9 hours to complete the task, [4 marks]
\item the range, symmetrical about the mean, within which 80% of the people's times lie. [5 marks]
\end{enumerate}
It is found that, in fact, 80% of the people take more than 5 hours. The model is modified so that the mean is still 8 hours but the standard deviation is no longer 2 hours.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the standard deviation of the times in the modified model. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q6 [12]}}