| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Probability calculation plus find unknown boundary |
| Difficulty | Moderate -0.8 This is a straightforward S1 normal distribution question requiring basic z-score calculations and inverse normal lookups. Part (a) is direct standardization and table reading, while part (b) requires finding a z-value from a given probability—both are routine textbook exercises with no problem-solving insight needed. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(P(X < 91) = P(Z < \frac{91 - 100}{15})\) | M1 | Attempt standardisation |
| \(= P(Z < -0.6)\) | A1 | |
| \(= 1 - 0.7257\) | M1 | |
| \(= 0.2743\) | A1 | awrt 0.274 |
| (b) \(1 - 0.2090 = 0.7910\) | B1 | 0.791 |
| \(P(X > 100 + k) = 0.2090\) or \(P(X < 100 + k) = 0.7910\) | M1 | (May be implied) |
| Use of tables to get \(z = 0.81\) | B1 | |
| \(\frac{100 + k - 100}{15} = 0.81\) | M1, A1 f.t | (ft their \(z = 0.81\), but must be \(z\) not prob.) |
| \(\mathbf{k = 12}\) | A1 cao | (6 marks) |
**(a)** $P(X < 91) = P(Z < \frac{91 - 100}{15})$ | M1 | Attempt standardisation |
$= P(Z < -0.6)$ | A1 |
$= 1 - 0.7257$ | M1 |
$= 0.2743$ | A1 | awrt **0.274** | (4 marks)
**(b)** $1 - 0.2090 = 0.7910$ | B1 | 0.791 |
$P(X > 100 + k) = 0.2090$ or $P(X < 100 + k) = 0.7910$ | M1 | (May be implied) |
Use of tables to get $z = 0.81$ | B1 |
$\frac{100 + k - 100}{15} = 0.81$ | M1, A1 f.t | (ft their $z = 0.81$, but must be $z$ not prob.) |
$\mathbf{k = 12}$ | A1 cao | (6 marks)
**Total: 10 marks**
---\begin{enumerate}
\item The measure of intelligence, IQ, of a group of students is assumed to be Normally distributed with mean 100 and standard deviation 15.\\
(a) Find the probability that a student selected at random has an IQ less than 91.
\end{enumerate}
The probability that a randomly selected student has an IQ of at least $100 + k$ is 0.2090 .\\
(b) Find, to the nearest integer, the value of $k$.\\
\hfill \mbox{\textit{Edexcel S1 2007 Q7 [10]}}