How to show for a simple regression with an intercept and one independent variable $R^2 = r ^2$ , where $r$ is the ordinary correlation coefficient. How to show for a simple regression with an intercept and one independent variable $R^2 = r ^2$, where $r$ is the ordinary correlation coefficient.
Here is where I'm at.
$R^2= \textrm{SSR}/\textrm{SST}$ 
then I substituted for $\hat{Y}$. 
Now my question is what is the difference and meaning between $\hat{Y}$, $\bar{Y}$, and $Y$ in the substitution of SSR and SST? 
 A: Consider:
The Pearson Product Moment Correlation Coefficient $r$ is an estimate of $\rho$, the population correlation coefficient, which measures the strength of a linear relationship between the two variables $x$ and $y$ ($x$ independent and $y$ dependent):
$r$ $=$ $\dfrac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i-\bar{x})^2 \cdot \sum_{i=1}^{n}(y_i-\bar{y})^2 }}$
where the bar in $\bar{x}$ and $\bar{y}$ represents the mean value of $x$ and $y$. $R^2$ is the percentage of variance accounted for by the regression model and is defined as indicated above and more fully as:
$R^2$ $=$ $\dfrac{Sum Squares Regression}{Sum Squares Total)}$ $=$ $\dfrac{\sum_{i=1}^{n}(\hat{y_i}-\bar{y})^2}{\sum_{i=1}^{n}(y_i-\bar{y})^2}$ 
where the hat in $\hat{y_i}$ represents the estimated $ith$ $y$ value using the estimated regression line. Note that $\hat{Y}$ represents the estimated $Y$ value based on the estimated regression line, $\bar{Y}$ represents the mean of the dependent variable $Y$, and $Y$ represents the dependent variable.
Post Script: $\hat{y_i}$ $=$ $b_0 + b_1 \cdot x_i$  where $b_0$ is the estimate of the intercept and $b_1$ is the estimate of the slope.
