| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Mixed calculations with boundaries |
| Difficulty | Standard +0.8 This question requires students to work backwards from percentiles to find distribution parameters using inverse normal tables, then apply the result to find another probability. Part (a) involves setting up and solving simultaneous equations with z-scores (z = -1.036 for 15th percentile, z = 1.645 for 95th percentile), requiring algebraic manipulation. Part (b) is more routine once parameters are known. The multi-step nature, need for inverse normal lookup, and algebraic solving elevate this above a standard S1 question but it remains a recognizable exam technique. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks |
|---|---|
| \(P(X < 60) = 0.15\) where \((60 - \mu)/\sigma = -1.04\) therefore \(60 - \mu = -1.04\sigma\) | M1 A1 |
| \(P(X > 90) = 0.05\) where \((90 - \mu)/\sigma = 1.65\) therefore \(90 - \mu = 1.65\sigma\) | M1 A1 |
| \(2.69\sigma = 30\) therefore \(\sigma = 11.2\), \(\mu = 71.6\) | M1 A1 A1 |
| (b) \(P(X > 80) = P(Z > 0.75) = 0.227\) therefore \(0.227 \times 200 = 45\) | M1 A1 A1 A1 |
**(a)**
$P(X < 60) = 0.15$ where $(60 - \mu)/\sigma = -1.04$ therefore $60 - \mu = -1.04\sigma$ | M1 A1 |
$P(X > 90) = 0.05$ where $(90 - \mu)/\sigma = 1.65$ therefore $90 - \mu = 1.65\sigma$ | M1 A1 |
$2.69\sigma = 30$ therefore $\sigma = 11.2$, $\mu = 71.6$ | M1 A1 A1 |
**(b)** $P(X > 80) = P(Z > 0.75) = 0.227$ therefore $0.227 \times 200 = 45$ | M1 A1 A1 A1 |
**Total marks: 10**
---The ages of the residents of a retirement community are assumed to be normally distributed. 15% of the residents are under 60 years old and 5% are over 90 years old.
\begin{enumerate}[label=(\alph*)]
\item Using this information, find the mean and the standard deviation of the ages. [7 marks]
\item If there are 200 residents, find how many are over 80 years old. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q5 [10]}}