| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Single normal population sample mean |
| Difficulty | Moderate -0.8 This is a straightforward application of the Central Limit Theorem requiring students to state the sampling distribution of the sample mean (normal with mean 62 and standard error 8.2/√50) and calculate a single probability using standardization. Both parts are routine textbook exercises with no problem-solving or novel insight required, making it easier than average but not trivial since it requires correct application of the CLT formula. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) (Approx) normal | B1 | |
| mean 62 | B1 | |
| \(sd = \frac{8.2}{\sqrt{50}} = 1.16\) (3 sfs) | B1 | or var \(= \frac{8.2^2}{50} = 1.34\) (3 sfs) |
| [3] | ||
| (ii) \(\frac{64-62}{\text{"1.16"}} (= 1.725 \text{ or } 1.724)\) | M1 | For standardising \(\div \sqrt{50}\) essential (no CC) |
| \(1 - \Phi(\text{"1.725"}) = (1 - 0.9577) = 0.0423\) (3 sfs) | M1 | For correct area consistent with their mean |
| A1 | [3] |
**(i)** (Approx) normal | B1 |
mean 62 | B1 |
$sd = \frac{8.2}{\sqrt{50}} = 1.16$ (3 sfs) | B1 | or var $= \frac{8.2^2}{50} = 1.34$ (3 sfs)
| [3] |
**(ii)** $\frac{64-62}{\text{"1.16"}} (= 1.725 \text{ or } 1.724)$ | M1 | For standardising $\div \sqrt{50}$ essential (no CC)
$1 - \Phi(\text{"1.725"}) = (1 - 0.9577) = 0.0423$ (3 sfs) | M1 | For correct area consistent with their mean
| A1 | [3]3 The weight, in grams, of a certain type of apple is modelled by the random variable $X$ with mean 62 and standard deviation 8.2. A random sample of 50 apples is selected, and the mean weight in grams, $\bar { X }$, is found.\\
(i) Describe fully the distribution of $\bar { X }$.\\
(ii) Find $\mathrm { P } ( \bar { X } > 64 )$.
\hfill \mbox{\textit{CAIE S2 2010 Q3 [6]}}