S3 June 2014 Q2
2. The weights of pears in an orchard are assumed to have unknown mean \(\mu\) and unknown standard deviation \(\sigma\).
A random sample of 20 pears is taken and their weights recorded.
The sample is represented by \(X_1, X_2, \ldots, X_{20}\). State whether or not the following are statistics. Give reasons for your answers.
(a)
(i) \(\dfrac{X_1 + 3X_{20}}{2}\)
(ii) \(\displaystyle\sum_{i=1}^{20} (X_i - \mu)\)
(iii) \(\displaystyle\sum_{i=1}^{20} \left(\frac{X_i - \mu}{\sigma}\right)\)
(4)(b) Find the mean and variance of \(\dfrac{3X_1 - X_{20}}{2}\) (3)
| Scheme | Marks |
|---|---|
| (i) Only contains known data / function of data only / no population parameters | B1 |
| therefore it is a statistic | B1d |
| (ii) and (iii) contain unknown parameters / population parameters / \(\mu\) and / or \(\sigma\) | B1 |
| therefore it is not a statistic. | B1d |
| (4) |
Notes
First B1 for known / no unknowns o.e. in (i)
Second B1 dependent on first B1 for ‘Yes’ / is a statistic o.e. in (i)
Third B1 for unknowns o.e. in both (ii) and (iii)
Fourth B1 dependent on third B1 for ‘No’ / not a statistic o.e. in both (ii) and (ii)
| Scheme | Marks |
|---|---|
| \(\left(\mathrm{E}\left(\dfrac{3X_1 - X_{20}}{2}\right) = \dfrac{3\mu - \mu}{2} =\right)\ \mu\) | B1 |
| \(\mathrm{Var}\left(\dfrac{3X_1 - X_{20}}{2}\right) = \dfrac{9\sigma^2 + \sigma^2}{2^2}\) | M1 |
| \(= \dfrac{5\sigma^2}{2}\) | A1 |
| (3) | |
| (7 marks) |
Notes
B1 for \(\mu\)
M1 for some squaring on numerator or denominator and must add on numerator
A1 for \(\dfrac{5\sigma^2}{2}\) o.e.