changed the position of the slope and independent term equations.

LinearModel/LinearModel.Rmd

@@ -43,7 +43,10 @@ The term "regression" was introduced by Francis Galton (Darwin's nephew) during
 
 The general equation for the straight line is $y = mx + b_0$, this form is the "slope, intersection form". The slope is the rate of change the gives the change in $y$ for a unit change in $x$. Remember that the slope formula for two pair of points $(x_1, y_1)$ and $(x_2, y_2)$ is:  
 $$ m = \frac{(y_2 - y_1)}{(x_2 - x_1)} $$  
-Using the expression for all the lines $y_i = \beta_{0_i} + \beta_{1_i}x_i$ for a set of points $(x_i, y_i)$, finding the minimum of the addition of all the differences with the ideal line we estimate the expression for the "best" slope $\hat{\beta_{1}}$ and the independent term $\hat{\beta_{0}}$:
+Using the expression for all the lines $y_i = \beta_{0_i} + \beta_{1_i}x_i$ for a set of points $(x_i, y_i)$, finding the minimum of the addition of all the differences with the ideal line we estimate the expression for the "best" slope $\hat{\beta_{1}}$ and the independent term $\hat{\beta_{0}}$:  
+
+$$\hat{\beta_{1}} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i -\bar{y} )}{\sum_{i=1}^{n}(x_i - \bar{x})^2} $$  
+$$\hat{\beta_{0}} = \bar{y} - \hat{\beta_{1}} \bar x $$  
 
 The basic properties we know about one variable linear regression are:  
   The correlation measures the strength of the relationship between x and y (see this shiny app for an excellent visual overview of correlations).  
@@ -52,9 +55,6 @@ The basic properties we know about one variable linear regression are:
   The slope of the line is defined as the change in $y$ over the change in $x$; $m= \frac{\Delta y}{\Delta x}$.  
 For regression use the ratio of the standard deviations such that the correlation is defined as $m=r\frac{s_y}{s_x}$ where $m$ is the slope, $r$ is the correlation and $\bar{x}$ and $\bar{y}$ the mean, and $s$ is the sample standard deviation.  
 
-$$\hat{\beta_{1}} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i -\bar{y} )}{\sum_{i=1}^{n}(x_i - \bar{x})^2} $$
-
-$$\hat{\beta_{0}} = \bar{y} - \hat{\beta_{1}} \bar x $$  
 
 ## Example of linear regression  
 

LinearModel/LinearModel.html

@@ -1578,6 +1578,11 @@ <h2>The Least-Square linear regression</h2>
 addition of all the differences with the ideal line we estimate the
 expression for the “best” slope <span class="math inline">\(\hat{\beta_{1}}\)</span> and the independent term
 <span class="math inline">\(\hat{\beta_{0}}\)</span>:</p>
+<p><span class="math display">\[\hat{\beta_{1}} = \frac{\sum_{i=1}^{n}
+(x_i - \bar{x})(y_i -\bar{y} )}{\sum_{i=1}^{n}(x_i - \bar{x})^2}
+\]</span><br />
+<span class="math display">\[\hat{\beta_{0}} = \bar{y} - \hat{\beta_{1}}
+\bar x \]</span></p>
 <p>The basic properties we know about one variable linear regression
 are:<br />
 The correlation measures the strength of the relationship between x and
@@ -1589,11 +1594,6 @@ <h2>The Least-Square linear regression</h2>
 \frac{\Delta y}{\Delta x}\)</span>.<br />
 For regression use the ratio of the standard deviations such that the
 correlation is defined as <span class="math inline">\(m=r\frac{s_y}{s_x}\)</span> where <span class="math inline">\(m\)</span> is the slope, <span class="math inline">\(r\)</span> is the correlation and <span class="math inline">\(\bar{x}\)</span> and <span class="math inline">\(\bar{y}\)</span> the mean, and <span class="math inline">\(s\)</span> is the sample standard deviation.</p>
-<p><span class="math display">\[\hat{\beta_{1}} = \frac{\sum_{i=1}^{n}
-(x_i - \bar{x})(y_i -\bar{y} )}{\sum_{i=1}^{n}(x_i - \bar{x})^2}
-\]</span></p>
-<p><span class="math display">\[\hat{\beta_{0}} = \bar{y} -
-\hat{\beta_{1}} \bar x \]</span></p>
 </div>
 <div id="example-of-linear-regression" class="section level2">
 <h2>Example of linear regression</h2>