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Show that the estimated regression equation can be written as:

$\hat{y} = \bar{y} + \hatβ_1(x - \bar{x})$

Say what this tells us about the fitted regression line.

Okay so I know $y_i = β_0 + β_1x_i$ but I'm not sure how to manipulate this to form the equation that is asked for above. I also don't really know what this would say about the line. $β_1$ is the slope yes? but what does the loss of $β_0$ to $\bar{y}$ and $x_i$ to $(x - \bar{x})$ mean?

Any help is greatly appreciated as this is stopping me from proceeding further into the question :'(

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First, when you solve the normal equations, you will find that $\hat{y}_{i}$ = $\hat{\beta}_{0}$+$\hat{\beta}_{1}x_{i}$. You will also find that $\bar{y}$ = $\hat{\beta}_{0}$+$\hat{\beta}_{1} \bar{x}$. Writing $\hat{\beta}_{0}$= $\bar{y}$ -$\hat{\beta}_{1} \bar{x}$ and inserting into the first equation should yield the result you want.

You are correct that the interpretation of $\beta_{1}$ as the slope is still correct. I think the point of the question is to inform the student that sometimes, people perform simple linear regression by centering the $x$ variable, i.e. by subtracting the mean of $x$ from each $x_{i}$ value.

The take home message is - performing a simple linear regression on a centered $x$ variable leads to the same slope estimate as the uncentered case but a different estimate of the intercept (it is now the average of the observed $y$'s).

A simple linear regression with a centered $x$ variable leads to least square estimators of the intercept and slope which are uncorrelated. There are also advantages to centering when you want to fit a polynomial regression. This helps reduce the collinearity between the variable and itself raised to some power.

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