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Pooled slope of groups

StatisticsHard

Groups A and B, with $10$ points each, are pooled into one data set of $20$ points. Group A has mean $x = 2$ , mean $y = 5$ , within-group $S_{xx} = 20$ and within-group $S_{x y} = 30$ . Group B has mean $x = 6$ , mean $y = 11$ , within-group $S_{xx} = 20$ and within-group $S_{x y} = 10$ . What is the least-squares slope of $y$ on $x$ for the pooled data?

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Hint

Total sums of squares equal the within-group parts plus a between-group part built from the group means.

Solution

The pooled means are $4$ for $x$ and $8$ for $y$ .
The between-group part of $S_{xx}$ is $10 \times 2^{2} + 10 \times 2^{2} = 80$ , so the total $S_{xx} = 20 + 20 + 80 = 120$ .
The between-group part of $S_{x y}$ is $10 \times (- 2) (- 3) + 10 \times 2 \times 3 = 120$ , so the total $S_{x y} = 30 + 10 + 120 = 160$ .
The pooled slope is $\frac{160}{120} = \frac{4}{3}$ .

Answer: $\frac{4}{3}$

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