S3 June 2006 Q2
2. A report on the health and nutrition of a population stated that the mean height of three-year old children is 90 cm and the standard deviation is 5 cm. A sample of 100 three-year old children was chosen from the population.
(a) Write down the approximate distribution of the sample mean height. Give a reason for your answer. (3)
(b) Hence find the probability that the sample mean height is at least 91 cm. (3)
| Scheme | Marks |
|---|---|
| \(\bar{X} \sim \mathrm{N}\left(90, \dfrac{5^2}{100}\right)\) i.e. \(\mathrm{N}(90, 0.25)\) | M1 A1 |
| Application of central limit theorem as (sample large) | B1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(\bar{X} \geqslant 91) = 1 - \mathrm{P}\left(Z \lt \dfrac{91 - 90}{0.5}\right)\) Stand. | M1 A1 |
| \(= 1 - \mathrm{P}(Z \lt 2)\) \(= 1 - 0.9772\) \(= 0.0228\) awrt 0.0228 | A1 |
| (3) | |
| (6 marks) |