| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Probability calculation plus find unknown boundary |
| Difficulty | Standard +0.3 This is a standard S1 normal distribution question requiring routine z-score calculations and inverse normal lookups. Part (i) is direct standardization, part (ii) uses percentiles with given mean/SD, and part (iii) requires solving simultaneous equations from two percentiles—all textbook techniques with no novel problem-solving required. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(>65) = P\left(z > \dfrac{65-61.4}{12.3}\right) = P(z > 0.2927)\) | M1 | Standardising no continuity correction, no square or square root, condone \(\pm\) standardisation formula |
| M1 | Correct area \((< 0.5)\) | |
| \(= 1 - 0.6153 = 0.385\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
\(P(<65) = 0.6153\) so \(P(| B1 |
| |
| \(z = 1.105\) | B1 | \(z = \pm 1.105\) seen or rounding to \(1.1\) |
| \(1.105 = \dfrac{k - 61.4}{12.3}\) | M1 | Standardising allow \(\pm\), cc, sq rt, sq. Need to see use of tables backwards so must be a \(z\)-value, not \(1-z\) value |
| \(k = 75.0\) | A1 | Answers which round to \(75.0\). Condone \(75\) if supported |
| Answer | Marks | Guidance |
|---|---|---|
| \(2.326 = \dfrac{97.2 - \mu}{\sigma}\) | B1 | \(\pm 2.326\) seen (Use of critical value) |
| \(-0.44 = \dfrac{55.2 - \mu}{\sigma}\) | B1 | \(\pm 0.44\) seen |
| M1 | An equation with a \(z\)-value, \(\mu\), \(\sigma\) and \(97.2\) or \(55.2\), allow \(\sqrt{\sigma}\) or \(\sigma^2\) | |
| M1 | Algebraic elimination \(\mu\) or \(\sigma\) from *their* two simultaneous equations | |
| \(\mu = 61.9\), \(\sigma = 15.2\) | A1 | Both correct answers |
## Question 7(i):
$P(>65) = P\left(z > \dfrac{65-61.4}{12.3}\right) = P(z > 0.2927)$ | M1 | Standardising no continuity correction, no square or square root, condone $\pm$ standardisation formula
| M1 | Correct area $(< 0.5)$
$= 1 - 0.6153 = 0.385$ | A1 |
**Total: 3 marks**
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## Question 7(ii):
$P(<65) = 0.6153$ so $P(<k) = 0.25 + 0.6153 = 0.8653$ | B1 |
$z = 1.105$ | B1 | $z = \pm 1.105$ seen or rounding to $1.1$
$1.105 = \dfrac{k - 61.4}{12.3}$ | M1 | Standardising allow $\pm$, cc, sq rt, sq. Need to see use of tables backwards so must be a $z$-value, not $1-z$ value
$k = 75.0$ | A1 | Answers which round to $75.0$. Condone $75$ if supported
**Total: 4 marks**
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## Question 7(iii):
$2.326 = \dfrac{97.2 - \mu}{\sigma}$ | B1 | $\pm 2.326$ seen (Use of critical value)
$-0.44 = \dfrac{55.2 - \mu}{\sigma}$ | B1 | $\pm 0.44$ seen
| M1 | An equation with a $z$-value, $\mu$, $\sigma$ and $97.2$ or $55.2$, allow $\sqrt{\sigma}$ or $\sigma^2$
| M1 | Algebraic elimination $\mu$ or $\sigma$ from *their* two simultaneous equations
$\mu = 61.9$, $\sigma = 15.2$ | A1 | Both correct answers
**Total: 5 marks**7 In Jimpuri the weights, in kilograms, of boys aged 16 years have a normal distribution with mean 61.4 and standard deviation 12.3.\\
(i) Find the probability that a randomly chosen boy aged 16 years in Jimpuri weighs more than 65 kilograms.\\
(ii) For boys aged 16 years in Jimpuri, $25 \%$ have a weight between 65 kilograms and $k$ kilograms, where $k$ is greater than 65 . Find $k$.\\
In Brigville the weights, in kilograms, of boys aged 16 years have a normal distribution. $99 \%$ of the boys weigh less than 97.2 kilograms and $33 \%$ of the boys weigh less than 55.2 kilograms.\\
(iii) Find the mean and standard deviation of the weights of boys aged 16 years in Brigville.\\
\hfill \mbox{\textit{CAIE S1 2017 Q7 [12]}}