S3 June 2005 Q2
2. A sample of size 5 is taken from a population that is normally distributed with mean 10 and standard deviation 3. Find the probability that the sample mean lies between 7 and 10. (6)
| Scheme | Marks |
|---|---|
| \(X \sim \mathrm{N}(10, 3^2) \quad \therefore \quad \bar{X} \sim \mathrm{N}\left(10, \tfrac{9}{5}\right)\) Can be implied 10; \(\tfrac{9}{5}\) | B1; B1 |
| \(\mathrm{P}(7 \leqslant \bar{X} \leqslant 10) = \mathrm{P}\left(\dfrac{7 - 10}{\sqrt{9/5}} \lt Z \lt 0\right)\) Standardising with 10 & their \(\sigma\) | M1 A1 |
| \(= \mathrm{P}(-2.236 \lt Z \lt 0)\) | |
| \(= \Phi(0) - \{1 - \Phi(2.24)\}\) | M1 (\(p \lt 0.5\)) |
| \(= \underline{0.4875}\) | A1 |
| (6) | |
| (6 marks) |