Standard +0.8 This is a pooled variance estimation problem requiring knowledge of the formula and careful algebraic manipulation of summary statistics. While the technique is standard in Further Maths Statistics, it involves multiple computational steps (calculating sample variances from raw data summaries, then combining them with appropriate weights) and requires precision. It's moderately harder than average A-level questions but not exceptionally challenging for Further Maths students.
6 The independent random variables \(X\) and \(Y\) have normal distributions with the same variance \(\sigma ^ { 2 }\). Samples of 5 observations of \(X\) and 10 observations of \(Y\) are made, and the results are summarised by \(\Sigma x = 15 , \Sigma x ^ { 2 } = 128 , \Sigma y = 36\) and \(\Sigma y ^ { 2 } = 980\). Find a pooled estimate of \(\sigma ^ { 2 }\).
\(5 \times 16.6\) or \(10 \times 85.04\) or \(83\) or \(850.4\)
Award A1 for one term in numerator
or \(4 \times 20.75\) or \(9 \times 94.5\)
A1
\(71.8\)
A1
Total: 3
## Question 6:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $((128 - 15^2/5) + (980 - 36^2/10))/13$ | M1 | Use standard formula for pooled estimate |
| $5 \times 16.6$ or $10 \times 85.04$ or $83$ or $850.4$ | | Award A1 for one term in numerator |
| or $4 \times 20.75$ or $9 \times 94.5$ | A1 | |
| $71.8$ | A1 | **Total: 3** |
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6 The independent random variables $X$ and $Y$ have normal distributions with the same variance $\sigma ^ { 2 }$. Samples of 5 observations of $X$ and 10 observations of $Y$ are made, and the results are summarised by $\Sigma x = 15 , \Sigma x ^ { 2 } = 128 , \Sigma y = 36$ and $\Sigma y ^ { 2 } = 980$. Find a pooled estimate of $\sigma ^ { 2 }$.
\hfill \mbox{\textit{CAIE FP2 2008 Q6 [3]}}