6 The independent random variables \(X\) and \(Y\) have distributions with the same variance \(\sigma ^ { 2 }\). Random samples of \(N\) observations of \(X\) and 10 observations of \(Y\) are taken, and the results are summarised by $$\Sigma x = 5 , \quad \Sigma x ^ { 2 } = 11 , \quad \Sigma y = 10 , \quad \Sigma y ^ { 2 } = 160 .$$ These data give a pooled estimate of 12 for \(\sigma ^ { 2 }\). Find \(N\).

## Question 6:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(11 - 5^2/N + 160 - 10^2/10)/(N + 10 - 2) = 12$ | M1 A1 | Equate pooled estimate of $\sigma^2$ to 12 |
| $12N^2 - 65N + 25 = 0$, $N = 5$ | M1 A1 | Formulate and solve relevant quadratic equation for $N$ |
| **Total: 4 marks** | | |

6 The independent random variables $X$ and $Y$ have distributions with the same variance $\sigma ^ { 2 }$. Random samples of $N$ observations of $X$ and 10 observations of $Y$ are taken, and the results are summarised by

$$\Sigma x = 5 , \quad \Sigma x ^ { 2 } = 11 , \quad \Sigma y = 10 , \quad \Sigma y ^ { 2 } = 160 .$$

These data give a pooled estimate of 12 for $\sigma ^ { 2 }$. Find $N$.

\hfill \mbox{\textit{CAIE FP2 2015 Q6 [4]}}