Fixed LATEX/ R typo from Regression Models 01_03

LATEX:
\sum_{i=1^n} => \sum_{i=1}^n
\beta_1 => \hat \beta_1

R Code:
Fixed missing `/`

07_RegressionModels/01_03_ols/index.Rmd

@@ -103,7 +103,7 @@ is the least squares line.
 * Consider forcing $\beta_0 = 0$ and thus $\hat \beta_0=0$; 
 that is, only considering lines through the origin
 * The solution works out to be
-$$\hat \beta_1 = \frac{\sum_{i=1^n} Y_i X_i}{\sum_{i=1}^n X_i^2}.$$
+$$\hat \beta_1 = \frac{\sum_{i=1}^n Y_i X_i}{\sum_{i=1}^n X_i^2}.$$
 
 ---
 ## Let's show it
@@ -123,7 +123,7 @@ $$\hat \beta_1 = \frac{\sum_{i=1^n} Y_i X_i}{\sum_{i=1}^n X_i^2}.$$
 ## Recapping what we know
 * If we define $\mu_i = \beta_0$ then $\hat \beta_0 = \bar Y$.
  * If we only look at horizontal lines, the least squares estimate of the intercept of that line is the average of the outcomes.
-* If we define $\mu_i = X_i \beta_1$ then $\hat \beta_1 = \frac{\sum_{i=1^n} Y_i X_i}{\sum_{i=1}^n X_i^2}$
+* If we define $\mu_i = X_i \beta_1$ then $\hat \beta_1 = \frac{\sum_{i=1}^n Y_i X_i}{\sum_{i=1}^n X_i^2}$
  * If we only look at lines through the origin, we get the estimated slope is the cross product of the X and Ys divided by the cross product of the Xs with themselves.
 * What about when $\mu_i = \beta_0 + \beta_1 X_i$? That is, we don't want to restrict ourselves to horizontal lines or lines through the origin.
 
@@ -132,7 +132,7 @@ $$\hat \beta_1 = \frac{\sum_{i=1^n} Y_i X_i}{\sum_{i=1}^n X_i^2}.$$
 $$\begin{align} \
 \sum_{i=1}^n (Y_i - \hat \mu_i) (\hat \mu_i - \mu_i) 
 = & \sum_{i=1}^n (Y_i - \hat\beta_0 - \hat\beta_1 X_i) (\hat \beta_0 + \hat \beta_1 X_i - \beta_0 - \beta_1 X_i) \\
-= & (\hat \beta_0 - \beta_0) \sum_{i=1}^n (Y_i - \hat\beta_0 - \hat \beta_1 X_i) + (\beta_1 - \beta_1)\sum_{i=1}^n (Y_i - \hat\beta_0 - \hat \beta_1 X_i)X_i\\
+= & (\hat \beta_0 - \beta_0) \sum_{i=1}^n (Y_i - \hat\beta_0 - \hat \beta_1 X_i) + (\hat \beta_1 - \beta_1)\sum_{i=1}^n (Y_i - \hat\beta_0 - \hat \beta_1 X_i)X_i\\
 \end{align} $$
 Note that 
 
@@ -228,7 +228,7 @@ abline(mean(y) - mean(x) * cor(y, x) * sd(y) / sd(x),
  sd(y) / sd(x) * cor(y, x), 
  lwd = 3, col = "red")
 abline(mean(y) - mean(x) * sd(y) / sd(x) / cor(y, x), 
- sd(y) cor(y, x) / sd(x), 
+ sd(y) / cor(y, x) / sd(x), 
  lwd = 3, col = "blue")
 abline(mean(y) - mean(x) * sd(y) / sd(x), 
  sd(y) / sd(x),