S4 June 2005 Q1
1. The random variable \(X\) has a \(\chi^2\)-distribution with 9 degrees of freedom.
(a) Find \(\mathrm{P}(2.088 \lt X \lt 19.023)\). (3)
The random variable \(Y\) follows an F-distribution with 12 and 5 degrees of freedom.
(b) Find the upper and lower 5% critical values for \(Y\). (3)
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X \gt 19.023) = 0.025\) or \(\mathrm{P}(X \lt 19.023) = 0.975\) \(\mathrm{P}(X \gt 2.088) = 0.990\) or \(\mathrm{P}(X \lt 2.088) = 0.010\) both | B1 |
| \(\therefore\ \mathrm{P}(2.088 \lt X \lt 19.023) = 0.990 - 0.025\) or \(0.975 - 0.010\) | M1 |
| \(= 0.965\) | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| Upper critical value of \(F_{12,5} = 4.68\) | B1 |
| Lower critical value of \(F_{12,5} = \dfrac{1}{F_{5,12}}\) | M1 |
| \(= \dfrac{1}{3.11} = 0.3215\ldots\) awrt 0.322 | A1 |
| (3) |