I've worked the slope all the way down to $sum [x_i(y_i – bar{y})] = hatbeta_1 sum[x_i(x_i – bar{x})]$
But I can not figure out how to show the steps for:
$sum[x_i(y_i – bar{y})] = sum(x_i – bar{x})(y_i – bar{y})$
and
$hat beta_1 sum[x_i(x_i – bar{x})] = hat beta_1 sum(x_i – bar{x})^2$
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Best Answer
Add and subtract and then look whether the unwanted term disappears:
$$sum[x_i(y_i-bar y)]=sum[(x_i-bar x+bar x)(y_i-bar y)]=sum[(x_i-bar x)(y_i-bar y)]+sum [bar x(y_i-bar y)].$$
Now
$$sum [bar x(y_i-bar y)]=bar xsum [y_i-bar y]=bar xleft[sum y_i-nbar yright]=0,$$
where $n$ is the number of the terms in the sum. Apply the same trick for the second equation.
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