S4 June 2013 (R) Q1
1.
(a) Find the value of the constant \(a\) such that \[\mathrm{P}(1.690 \lt \chi^2_7 \lt a) = 0.95\] (2)
The random variable \(Y\) follows an \(F\)-distribution with 6 and 4 degrees of freedom.
(b)
(i) Find the upper 1% critical value for \(Y\).
(ii) Find the lower 1% critical value for \(Y\). (2)
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X \gt 1.690) = 0.975\) | |
| \(\mathrm{P}(X \gt a) = 0.025\) | M1 |
| \(a = 16.013\) | A1 |
| (2) |
Notes
M1 for using 0.025
| Scheme | Marks |
|---|---|
| Upper critical value of \(\mathrm{F}_{6,4} = 15.21\) | B1 |
| Lower critical value of \(\mathrm{F}_{6,4} = \dfrac{1}{9.15} = 0.109\) | B1 |
| (2) | |
| (4 marks) |
Notes
2nd B1 either \(\dfrac{1}{9.15}\) or awrt 0.109