| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Linear relationship μ = kσ |
| Difficulty | Standard +0.3 Part (a) requires setting up μ=2σ, standardizing to find z-score from tables (z≈-1.28), then solving a linear equation—straightforward application of normal distribution with one extra algebraic step. Part (b) is direct recall that P(μ-σ < X < μ+σ) ≈ 0.68, multiplied by 800. Both parts are routine S1 exercises requiring standard techniques without problem-solving insight. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\dfrac{5.2 - 2s}{s} = -1.282\) | M1 | Equation with \(\pm\) correct LHS seen here or later, can be \(\mu\) or \(s\), no cc |
| B1 | \(\pm 1.282\) seen, accept \(\pm 1.28\) or anything in between | |
| M1 | Solving their equation with recognisable \(z\)-value and only 1 unknown occurring twice | |
| \(s = 7.24\) or \(7.23\) | A1 4 | Correct final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\Phi\!\left(\dfrac{\mu + \sigma - \mu}{\sigma}\right) = 0.8413\) | B1 | \(0.8413\) (p) seen or implied (can use their own numbers) |
| \(P(\lvert z \rvert < 1) = 0.3413 \times 2 = 0.6826\) | M1 | Finding the correct area i.e. \(2p - 1\) |
| \(0.6826 \times 800 = 546\) (accept 547) | A1 3 | Correct answer, must be a positive integer |
| OR \(SR\ 800 \times 2/3 = 533\) or \(534\) | SR B1 | for \(2/3\) |
| B1 | for 533 or 534 or B2 if 533 or 534 and no working |
## Question 3:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\dfrac{5.2 - 2s}{s} = -1.282$ | M1 | Equation with $\pm$ correct LHS seen here or later, can be $\mu$ or $s$, no cc |
| | B1 | $\pm 1.282$ seen, accept $\pm 1.28$ or anything in between |
| | M1 | Solving their equation with recognisable $z$-value and only 1 unknown occurring twice |
| $s = 7.24$ or $7.23$ | A1 **4** | Correct final answer |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\Phi\!\left(\dfrac{\mu + \sigma - \mu}{\sigma}\right) = 0.8413$ | B1 | $0.8413$ (p) seen or implied (can use their own numbers) |
| $P(\lvert z \rvert < 1) = 0.3413 \times 2 = 0.6826$ | M1 | Finding the correct area i.e. $2p - 1$ |
| $0.6826 \times 800 = 546$ (accept 547) | A1 **3** | Correct answer, must be a positive integer |
| OR $SR\ 800 \times 2/3 = 533$ or $534$ | SR B1 | for $2/3$ |
| | B1 | for 533 or 534 or B2 if 533 or 534 and no working |
---3
\begin{enumerate}[label=(\alph*)]
\item The random variable $X$ is normally distributed. The mean is twice the standard deviation. It is given that $\mathrm { P } ( X > 5.2 ) = 0.9$. Find the standard deviation.
\item A normal distribution has mean $\mu$ and standard deviation $\sigma$. If 800 observations are taken from this distribution, how many would you expect to be between $\mu - \sigma$ and $\mu + \sigma$ ?
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2007 Q3 [7]}}