The independent variables \(X\) and \(Y\) have distributions with the same variance \(\sigma^2\). Random samples of \(N\) observations of \(X\) and \(2N\) observations of \(Y\) are taken, and the results are summarised by $$\Sigma x = 4, \quad \Sigma x^2 = 10, \quad \Sigma y = 8, \quad \Sigma y^2 = 102.$$ These data give a pooled estimate of \(10\) for \(\sigma^2\). Find \(N\). [5]

Question 6:
6 | (10 – 42/N + 102 – 82/2N) / (N + 2N – 2) (AEF) | M1 A1 | State or find expression for pooled estimate of σ2
(confusing biased/unbiased estimates may still earn
M1)
112 – 48/N = 10 (3N – 2), 15N 2 – 66N + 24 = 0 | M1 A1 | Equate to 10 and rearrange as quadratic
N = (66 ± 54) / 30 = 4 | A1 | Solve quadratic for N, rejecting root 0⋅4
Total: | 5
Question | Answer | Marks | Guidance

The independent variables $X$ and $Y$ have distributions with the same variance $\sigma^2$. Random samples of $N$ observations of $X$ and $2N$ observations of $Y$ are taken, and the results are summarised by
$$\Sigma x = 4, \quad \Sigma x^2 = 10, \quad \Sigma y = 8, \quad \Sigma y^2 = 102.$$

These data give a pooled estimate of $10$ for $\sigma^2$. Find $N$. [5]

\hfill \mbox{\textit{CAIE FP2 2017 Q6 [5]}}