| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2015 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Distribution of sample mean |
| Difficulty | Easy -1.2 This is a straightforward application of the sampling distribution of the mean from a normal population. Part (i) requires recalling that the sample mean is normally distributed with mean μ and variance σ²/n. Part (ii) is a routine standardization and normal table lookup. Both parts are direct recall with minimal calculation, making this easier than average. |
| Spec | 5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(N(352, \ldots)\) | B1 | No recovery in (ii) for each B mark |
| Variance \(= 2.9\) | B1 [2] | Accept \(sd = \sqrt{2.9} = 1.70(29)\) stated |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{354 - 352}{\sqrt{2.9}}\) \((= 1.174)\) | M1 | With their mean and var; or \(\frac{354.05 - 352}{\sqrt{2.9}}\) or correct restart \((= 1.204)\) |
| \(1 - \Phi(\text{'1.174'})\) | M1 | Accept sd/var mix; \(-\Phi(\text{'1.204'}) = 0.114\) (3 sf) |
| \(= 0.120\) (3 sf) | A1 [3] | Incorrect cc can score M1M1A0 |
# Question 1:
## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $N(352, \ldots)$ | B1 | No recovery in (ii) for each B mark |
| Variance $= 2.9$ | B1 [2] | Accept $sd = \sqrt{2.9} = 1.70(29)$ stated |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{354 - 352}{\sqrt{2.9}}$ $(= 1.174)$ | M1 | With their mean and var; or $\frac{354.05 - 352}{\sqrt{2.9}}$ or correct restart $(= 1.204)$ |
| $1 - \Phi(\text{'1.174'})$ | M1 | Accept sd/var mix; $-\Phi(\text{'1.204'}) = 0.114$ (3 sf) |
| $= 0.120$ (3 sf) | A1 [3] | Incorrect cc can score M1M1A0 |
---1 It is known that the number, $N$, of words contained in the leading article each day in a certain newspaper can be modelled by a normal distribution with mean 352 and variance 29. A researcher takes a random sample of 10 leading articles and finds the sample mean, $\bar { N }$, of $N$.\\
(i) State the distribution of $\bar { N }$, giving the values of any parameters.\\
(ii) Find $\mathrm { P } ( \bar { N } > 354 )$.
\hfill \mbox{\textit{CAIE S2 2015 Q1 [5]}}