Challenging +1.2 This is an S4 question requiring knowledge of pooled variance and chi-squared distribution for confidence intervals. While it involves multiple samples and the pooled estimator formula, the procedure is relatively standard once the correct approach is identified. The calculation requires combining sample variances and applying the chi-squared confidence interval formula, which is more advanced than typical AS-level content but follows a learned template for Further Maths S4 students.
Two independent random samples \(X_1, X_2, ..., X_n\) 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.
$$s_x = 14.67 \quad s_y = 12.07$$
Find the 99\% confidence interval for \(\sigma^2\) based on these two samples.
[5]
Two independent random samples $X_1, X_2, ..., X_n$ 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.
$$s_x = 14.67 \quad s_y = 12.07$$
Find the 99\% confidence interval for $\sigma^2$ based on these two samples.
[5]
\hfill \mbox{\textit{Edexcel S4 Q2 [5]}}