Standard +0.8 This is a Further Maths statistics question requiring knowledge of pooled variance estimation formula and algebraic manipulation. While the concept is standard for FM students, it requires careful setup of the pooled variance formula with different sample sizes (n and 2n), substitution of given summations, and solving a resulting equation. The multi-step algebraic work and the specific FM context place it moderately above average difficulty.
7 Two independent random variables \(X\) and \(Y\) have distributions with the same variance \(\sigma ^ { 2 }\). Random samples of \(n\) observations of \(X\) and \(2 n\) observations of \(Y\) are taken and the results are summarised by
$$\Sigma x = 10.0 , \quad \Sigma x ^ { 2 } = 25.0 , \quad \Sigma y = 15.0 , \quad \Sigma y ^ { 2 } = 43.5 .$$
Given that the pooled estimate of \(\sigma ^ { 2 }\) is 2 , find the value of \(n\).
7 Two independent random variables $X$ and $Y$ have distributions with the same variance $\sigma ^ { 2 }$. Random samples of $n$ observations of $X$ and $2 n$ observations of $Y$ are taken and the results are summarised by
$$\Sigma x = 10.0 , \quad \Sigma x ^ { 2 } = 25.0 , \quad \Sigma y = 15.0 , \quad \Sigma y ^ { 2 } = 43.5 .$$
Given that the pooled estimate of $\sigma ^ { 2 }$ is 2 , find the value of $n$.
\hfill \mbox{\textit{CAIE FP2 2013 Q7 [7]}}