| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Expected frequency with unknown parameter |
| Difficulty | Standard +0.3 This is a straightforward normal distribution problem requiring standard inverse normal table lookups and probability calculations. Part (a) involves finding σ from P(X < 16) using z-tables (routine procedure), and part (b) requires calculating P(X > 20) then multiplying by 75. Both parts follow standard S1 procedures with no conceptual challenges, though the two-part structure and need for careful table work places it slightly above average difficulty. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(P(X < 16) = P(Z < -1/\sigma)\) where \(-0.33 = -1/\sigma\) so \(\sigma = 3.03\) | M1 A1 M1 A1 | |
| (b) \(P(X > 20) = P(Z > 3/3.03) = P(Z > 0.99) = 1 - 0.839 = 0.161\), so would expect 12 to be over 20 | M1 A1 M1 A1 A1 | Total: 10 marks |
(a) $P(X < 16) = P(Z < -1/\sigma)$ where $-0.33 = -1/\sigma$ so $\sigma = 3.03$ | M1 A1 M1 A1 |
(b) $P(X > 20) = P(Z > 3/3.03) = P(Z > 0.99) = 1 - 0.839 = 0.161$, so would expect 12 to be over 20 | M1 A1 M1 A1 A1 | **Total: 10 marks**The random variable $X$ is normally distributed with mean 17. The probability that $X$ is less than 16 is 0.3707.
\begin{enumerate}[label=(\alph*)]
\item Calculate the standard deviation of $X$. [4 marks]
\item In 75 independent observations of $X$, how many would you expect to be greater than 20? [6 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q4 [10]}}