summary(cars)

##      speed           dist       
##  Min.   : 4.0   Min.   :  2.00  
##  1st Qu.:12.0   1st Qu.: 26.00  
##  Median :15.0   Median : 36.00  
##  Mean   :15.4   Mean   : 42.98  
##  3rd Qu.:19.0   3rd Qu.: 56.00  
##  Max.   :25.0   Max.   :120.00

head(cars)

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

First, let’s visualize the relationship between car speeds and stopping distances.

## `geom_smooth()` using formula = 'y ~ x'

Now, let’s fit a simple linear regression model to our data.

## 
## 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

Let’s fit a simple linear regression model using the lm() function in R. This model will help us understand how the stopping distance of cars can be predicted by their speed.

# Fit the linear model
cars_lm <- lm(dist ~ speed, data = cars)

# Display the summary of the model to interpret coefficients
summary(cars_lm)

## 
## 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

Understanding residuals is key to diagnosing the model. Let’s visualize them.

Let’s create an interactive 3D plot using plotly to visualize speed, distance, and car index in the dataset.

Simple linear regression models the relationship between two variables by fitting a linear equation to observed data. The equation is given by:

\[ y = \beta_0 + \beta_1x + \epsilon \]

where: - \(y\) is the dependent variable, - \(x\) is the independent variable, - \(\beta_0\) is the intercept of the regression line, - \(\beta_1\) is the slope of the regression line, and - \(\epsilon\) is the error term.

Therefore, Simple linear regression on the cars dataset suggests a strong linear relationship between car speed and stopping distance.

The coefficients \(\beta_0\) and \(\beta_1\) provide us with important information:

• \(\beta_0\) (Intercept): The expected value of \(y\) when \(x\) is 0.
• \(\beta_1\) (Slope): The change in the expected value of \(y\) for a one-unit change in \(x\).

The error term \(\epsilon\) represents the difference between the observed values and the values predicted by the linear equation.

Thank you for attending the presentation.