# Interpreting simple linear regression

Good day,

I will am currently doing some self-study on Simple Linear Regression. I understand the formula:

$$\hat{Y}_i=\beta_0+\beta_1 X_i$$

But what does it mean when the value of $\beta_1$ is 0? Does it mean that $X$ and $Y$ have no relationship?

• Since you have $\hat y_i$ on the left, your right hand side should have $\hat \beta$'s. While you're editing, please remove the "e" from the end of the word 'formula'; you have only one formula so you should not use the plural. – Glen_b May 3 '14 at 10:32
If $X$ and $Y$ have no linear relationship, i.e. $\beta_1=0$, then no matter what value $X$ is, the predicted value for $Y$, $\hat{Y}$, is $\beta_0$. If you draw the regression line, it would be a flat line with the y-intercept at $\beta_0$. It might help to also think about the interpretation of the slope $\beta_1$ as the change in $\hat{Y}$ for a 1-unit change in $X$. Notice that if the slope is 0, then the value of $\hat{Y}$ does not change with $X$ (no linear relationship).
There are also some U and inverse U shaped relationships. For example, the relationship between exam score and anxiety is inverse U shaped. $\beta_1$ isn't 0 in this case, but it doesn't reflect the relationship.