| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2002 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Probability calculation plus find unknown boundary |
| Difficulty | Moderate -0.3 This is a straightforward application of normal distribution with standard procedures: (a) uses inverse normal tables to find μ from a given percentage, (b) applies standardization and table lookup, (c) uses basic probability multiplication. All parts follow textbook methods with no novel problem-solving required, making it slightly easier than average for A-level. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks |
|---|---|
| \(P(L > 50.98) = 0.025\) | B1 |
| \(P\left(Z > \frac{50.98 - \mu}{0.5}\right) = 0.025\) | M1 A1 |
| \(\frac{50.98 - \mu}{0.5} = 1.96\) | M1 A1 |
| \(\mu = 50\) (*) | (5 marks) |
| Answer | Marks |
|---|---|
| \(P(49.25 < L < 50.75) = P\left(\frac{49.25 - 50}{0.5} < Z < \frac{50.75 - 50}{0.5}\right)\) | M1 |
| \(= P(-1.5 < Z < 1.5)\) | A1 |
| \(= 2\Phi(1.5) - 1\) | M1 |
| \(= 0.8664\) | A1 |
| (4 marks) |
| Answer | Marks |
|---|---|
| \(P(\text{Both}) = (1 - 0.8664)^2\) | M1 A1 |
| \(= 0.01784\ldots\) | A1 |
| (2 marks) |
## (a)
$P(L > 50.98) = 0.025$ | B1 |
$P\left(Z > \frac{50.98 - \mu}{0.5}\right) = 0.025$ | M1 A1 |
$\frac{50.98 - \mu}{0.5} = 1.96$ | M1 A1 |
$\mu = 50$ (*) | (5 marks) |
## (b)
$P(49.25 < L < 50.75) = P\left(\frac{49.25 - 50}{0.5} < Z < \frac{50.75 - 50}{0.5}\right)$ | M1 |
$= P(-1.5 < Z < 1.5)$ | A1 |
$= 2\Phi(1.5) - 1$ | M1 |
$= 0.8664$ | A1 |
| (4 marks) |
## (c)
$P(\text{Both}) = (1 - 0.8664)^2$ | M1 A1 |
$= 0.01784\ldots$ | A1 |
| (2 marks) |
**Total: 11 marks**
---Strips of metal are cut to length $L$ cm, where $L \sim N(\mu, 0.5^2)$.
\begin{enumerate}[label=(\alph*)]
\item Given that 2.5\% of the cut lengths exceed 50.98 cm, show that $\mu = 50$. [5]
\item Find $P(49.25 < L < 50.75)$. [4]
\end{enumerate}
Those strips with length either less than 49.25 cm or greater than 50.75 cm cannot be used.
Two strips of metal are selected at random.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the probability that both strips cannot be used. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2002 Q4 [11]}}