Here is the corresponding general SLR Regression Model Equation:

\[ y = \beta_0 + \beta_1 x + \varepsilon \] This formula is what formalizes the empirical model!

We know that:

• \(y\) is our response variable
• \(x\) is the predictor variable
• \(\beta_0\) and \(\beta_1\) are parameters of the function that happen to be unknown at first
• \(\varepsilon\) is the random error

The coefficients are those unknown parameters defined in the SLR Equation, \(\beta_0\) and \(\beta_1\), respectively. With these coefficients, we are actually able to estimate them using something called the OLS method, or Ordinary Least Squares. What OLS does it is figures out a particular line that, effectively, minimizes what is known as the sum of squared vertical distances between the actual fitted line and the observed points. (This is what we also call residuals!) This gives us four equations to use:

• Residuals: \(e_{i} = Y_{i} - \hat{Y_{i}}\)
• Method of Least Squares: \(\sum_{i=1}^{n} (Y_{i} - \hat{\beta_0} - \hat{\beta_1}X_{i})^2\)
• \(\hat{\beta_1} = \frac{\sum (X_{i} - \bar X)(Y_{i} - \bar Y)}{\sum (X_{i} - \bar X)^2}\)
• \(\hat{\beta_0} = \bar Y - \hat{\beta_1} \bar X\)

With simple linear regression in mind, the \(\textbf{cars}\) dataset is a spectacular choice that is built into R that can be observed.

# Shows the first couple observations of the cars dataset
head(cars)

##   speed dist
## 1     4    2
## 2     4   10
## 3     7    4
## 4     7   22
## 5     8   16
## 6     9   10

nrow(cars)

## [1] 50

summary(fittedModel)

## 
## Call:
## lm(formula = dist ~ speed, data = cars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -29.069  -9.525  -2.272   9.215  43.201 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -17.5791     6.7584  -2.601   0.0123 *  
## speed         3.9324     0.4155   9.464 1.49e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.38 on 48 degrees of freedom
## Multiple R-squared:  0.6511, Adjusted R-squared:  0.6438 
## F-statistic: 89.57 on 1 and 48 DF,  p-value: 1.49e-12

After using the summary function on the fitted model of the cars dataset, it becomes apparent that we can then interpret some of the unknown values that were described in the original formula:

• We can see that \(\beta_0\), or the intercept, is equal to -17.579. This indicates that when our speed is nothing, or 0 mph, the stopping distance is that specific value. (According to the formula)
• We can also see that \(\beta_1\), or the slope, is equal to 3.932. This indicates that when we increase in speed, we are increasing by 3.932 feet for every increase in mph.
• Some small things that we can also see are the \(R^2 = 0.6511\) and p-value = 1.49e-12
• The R-Squared value simply tells us that roughly 65% of the variability in the stopping distance is caused strictly by speed (with other factors being the rest)
• The p-value in the summary is there to answer the question of whether the independent variable is actually significant in predicting the dependent variable. (Typical alpha values are set to a value of 0.05, so if the resulting p-value is less than that, then the independent variable is significant)