S4 June 2011 Q2
2. Two independent random samples \(X_1, X_2, \ldots, X_7\) and \(Y_1, Y_2, Y_3, Y_4\) were taken from different normal populations with a common standard deviation \(\sigma\).
The following sample statistics were calculated.
\[s_x = 14.67 \qquad s_y = 12.07\]Find the 99% confidence interval for \(\sigma^2\) based on these two samples. (5)
| Scheme | Marks |
|---|---|
| \(s_p^{\,2} = \dfrac{6s_x^{\,2} + 3s_y^{\,2}}{9} \quad (= 192.03\ldots)\) | M1 |
| \(1.735 \lt \dfrac{9s_p^{\,2}}{\sigma^2} \lt 23.589\) | B1M1B1 |
| So 99% confidence interval is \((73.26\ldots,\ 996.14\ldots)\) awrt (73.3, 996) | A1 |
| (5 marks) |
Notes
1st M1 for attempting \(s_p^{\,2}\)
1st B1 for 1.735 (or better)
2nd M1 for use of \(\dfrac{9s_p^{\,2}}{\sigma^2}\), follow through their \(s_p^{\,2}\)
2nd B1 for 23.589 (or better)
A1 for both values correct to awrt 3 sf