| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2018 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Estimate from percentile/frequency data |
| Difficulty | Standard +0.8 Part (a) is routine standardization and table lookup. Part (b) requires setting up and solving simultaneous equations from two percentiles (10.5th and 14.5th percentiles), involving inverse normal calculations and algebraic manipulation—significantly more demanding than standard normal distribution questions but still within typical A-level scope. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(X < 29.4) = P\!\left(Z < \dfrac{29.4 - 31.4}{\sqrt{3.6}}\right) = P(Z < -1.0541)\) | M1 | Standardise, no cc, must have square root |
| \(= 1 - 0.8540\) | M1 | Obtain \(1 - \text{prob}\) |
| \(= 0.146\) | A1 | Correct final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(X < 12) = \frac{42}{400} = 0.105\) and \(P(X > 19) = \frac{58}{400} = 0.145\) | M1 | Equation with \(\mu, \sigma\) and a \(z\)-value. Allow cc, wrong sign, but not \(\sqrt{\sigma}\) or \(\sigma^2\) |
| \(\dfrac{12 - \mu}{\sigma} = -1.253\) | B1 | Any form with \(z\) value rounding to \(\pm 1.25\) |
| \(\dfrac{19 - \mu}{\sigma} = 1.058\) | B1 | Any form with \(z\) value rounding to \(\pm 1.06\) |
| \(12 - \mu = -1.253\sigma\); \(19 - \mu = 1.058\sigma\); \(7 = 2.307\sigma\) or \(36.455 + 2.307\mu = 0\) oe | M1 | Solve 2 equations in \(\mu\), \(\sigma\) eliminating to 1 unknown |
| \(\mu = 15.8,\ \sigma = 3.03\) | A1 | Correct answers |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(X < 29.4) = P\!\left(Z < \dfrac{29.4 - 31.4}{\sqrt{3.6}}\right) = P(Z < -1.0541)$ | M1 | Standardise, no cc, must have square root |
| $= 1 - 0.8540$ | M1 | Obtain $1 - \text{prob}$ |
| $= 0.146$ | A1 | Correct final answer |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(X < 12) = \frac{42}{400} = 0.105$ and $P(X > 19) = \frac{58}{400} = 0.145$ | M1 | Equation with $\mu, \sigma$ and a $z$-value. Allow cc, wrong sign, but not $\sqrt{\sigma}$ or $\sigma^2$ |
| $\dfrac{12 - \mu}{\sigma} = -1.253$ | B1 | Any form with $z$ value rounding to $\pm 1.25$ |
| $\dfrac{19 - \mu}{\sigma} = 1.058$ | B1 | Any form with $z$ value rounding to $\pm 1.06$ |
| $12 - \mu = -1.253\sigma$; $19 - \mu = 1.058\sigma$; $7 = 2.307\sigma$ or $36.455 + 2.307\mu = 0$ oe | M1 | Solve 2 equations in $\mu$, $\sigma$ eliminating to 1 unknown |
| $\mu = 15.8,\ \sigma = 3.03$ | A1 | Correct answers |4
\begin{enumerate}[label=(\alph*)]
\item It is given that $X \sim \mathrm {~N} ( 31.4,3.6 )$. Find the probability that a randomly chosen value of $X$ is less than 29.4.
\item The lengths of fish of a particular species are modelled by a normal distribution. A scientist measures the lengths of 400 randomly chosen fish of this species. He finds that 42 fish are less than 12 cm long and 58 are more than 19 cm long. Find estimates for the mean and standard deviation of the lengths of fish of this species.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2018 Q4 [8]}}