6 The independent random variables \(X\) and \(Y\) have distributions with the same variance \(\sigma ^ { 2 }\). Random samples of 5 observations of \(X\) and \(n\) observations of \(Y\) are made and the results are summarised by $$\Sigma x = 5.5 , \quad \Sigma x ^ { 2 } = 15.05 , \quad \Sigma y = 8.0 , \quad \Sigma y ^ { 2 } = 36.4$$ Given that the pooled estimate of \(\sigma ^ { 2 }\) is 3 , find the value of \(n\).

## Question 6:

| Working/Answer | Mark | Guidance |
|---|---|---|
| Find pooled estimate: $\frac{(15{\cdot}05 - 5{\cdot}5^2/5 + 36{\cdot}4 - 8^2/n)}{(3+n)}$ | M1 A1 | |
| Equate to 3 and rearrange: $45{\cdot}4 - 64/n = 9 + 3n$ | M1 A1 | |
| $3n^2 - 36{\cdot}4n + 64 = 0$ | A1 | |
| Solve for $n$: $n = (36{\cdot}4 \pm 23{\cdot}6)/6 = 10$ | M1 A1 | [Subtotal: 7] |

**Total: 7**

---

6 The independent random variables $X$ and $Y$ have distributions with the same variance $\sigma ^ { 2 }$. Random samples of 5 observations of $X$ and $n$ observations of $Y$ are made and the results are summarised by

$$\Sigma x = 5.5 , \quad \Sigma x ^ { 2 } = 15.05 , \quad \Sigma y = 8.0 , \quad \Sigma y ^ { 2 } = 36.4$$

Given that the pooled estimate of $\sigma ^ { 2 }$ is 3 , find the value of $n$.

\hfill \mbox{\textit{CAIE FP2 2011 Q6 [7]}}