Hey, PF(adsbygoogle = window.adsbygoogle || []).push({});

I'm reading the following derivation of least squares, and I'm trying to figure out how the author went from the last step at the bottom of pg. 7 to the final equation (11) at the top of pg. 8.

[Harvard.edu]

More specifically, why is the denominator a difference of two terms? Aren't the terms in the denominator summed in the prior step?

I would expect the answer to be

$$

b_1=\dfrac{\displaystyle \sum_{\textrm{i=1}}^{n}y_ix_{i}-\left(\frac{1}{n}\right)\left(\sum_{\textrm{i=1}}^{n}y_i\sum_{\textrm{i=1}}^{n}x_{i}\right)}{\displaystyle\sum_{\textrm{i=1}}^{n}x_{i}^2+\left(\frac{1}{n}\right)\left(\sum_{\textrm{i=1}}^{n}x_{i}\right)^{2}}

$$

Note: I'm no statistician, but I thought you guys might be more familiar with this derivation.

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# Least Squares Derivation—Simple Algebraic Simplification

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