S3 June 2013 (R) Q2
2. A random sample of size \(n\) is to be taken from a population that is normally distributed with mean 40 and standard deviation 3. Find the minimum sample size such that the probability of the sample mean being greater than 42 is less than 5%. (5)
| Scheme | Marks |
|---|---|
| \(X \sim \mathrm{N}(40, 3^2) \quad \bar{X} \sim \mathrm{N}\left(40, \dfrac{9}{n}\right)\) (Condone \(Y \sim \mathrm{N}\left(40, \dfrac{9}{n}\right)\)) | B1 |
| \(\mathrm{P}(\bar{X} \gt 42) = \mathrm{P}\left(Z \gt \dfrac{42 - 40}{\sqrt{\frac{9}{n}}}\right)\) | M1 |
| \(\dfrac{42 - 40}{\sqrt{\frac{9}{n}}} \geqslant 1.6449\) | B1 dM1 |
| \(n \geqslant 6.087\) \(n = 7\) | A1 |
| (5 marks) |
Notes
1st B1 for stating the correct distribution for \(\bar{X}\).
May be implied if correctly used in line 2 and no incorrect version seen elsewhere.
1st M1 for an attempt to standardise with 42, 40 and their \(\sqrt{\dfrac{9}{n}}\), must have \(n\). Allow \(\pm\)
2nd B1 for using \(z = \pm 1.6449\) (or better)
2nd dM1 Dep on 1st M1 for forming an equation in \(n\) or \(\sqrt{n}\). Allow “=” or “<”
i.e. setting their standardised expression = their \(z\) value (\(|z| \gt 1.5\))
A1 for \(n = 7\) only
The A1 must follow from correct working so e.g. \(n \lt 6.087\) leading to \(n = 7\) is A0