| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Mixed calculations with boundaries |
| Difficulty | Standard +0.8 This question requires setting up and solving simultaneous equations using inverse normal distribution (z-scores) from two given percentiles, which goes beyond routine normal distribution problems. Students must recognize that P(X > 200) = 0.01 gives one equation and P(165 < X < 200) = 0.76 combined with the first condition gives another, then solve algebraically for μ and σ². The multi-step algebraic manipulation and conceptual understanding of how percentiles relate to parameters makes this moderately challenging for S1 level. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(P(X > 200) = 0.01\): \(\frac{200 - \mu}{\sigma} = 2.33\) → \(200 - \mu = 2.33\sigma\) | M1 A1 | \(P(X > 165) = 0.77\): \(\frac{165 - \mu}{\sigma} = -0.74\) → \(165 - \mu = -0.74\sigma\) |
| (b) \(P(X < 158) = P(Z < -1.34) = 1 - 0.91\), so 99% are under 158 cm | M1 A1 A1 | |
| (c) Model may be affected by lack of very short individuals (e.g. children), etc. | B2 | Total: 14 marks |
(a) $P(X > 200) = 0.01$: $\frac{200 - \mu}{\sigma} = 2.33$ → $200 - \mu = 2.33\sigma$ | M1 A1 | $P(X > 165) = 0.77$: $\frac{165 - \mu}{\sigma} = -0.74$ → $165 - \mu = -0.74\sigma$ | B1 M1 A1 | $3.07\sigma = 35$ → $\sigma = 11.4$ | M1 A1 A1 A1 | $\mu = 173$, $\sigma^2 = 130$ |
(b) $P(X < 158) = P(Z < -1.34) = 1 - 0.91$, so 99% are under 158 cm | M1 A1 A1 |
(c) Model may be affected by lack of very short individuals (e.g. children), etc. | B2 | Total: 14 marks
---The heights of the students at a university are assumed to follow a normal distribution. 1% of the students are over 200 cm tall and 76% are between 165 cm and 200 cm tall.
Find
\begin{enumerate}[label=(\alph*)]
\item the mean and the variance of the distribution, [9 marks]
\item the percentage of the students who are under 158 cm tall. [3 marks]
\item Comment briefly on the suitability of a normal distribution to model such a population. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q4 [14]}}