Linear regressions are used to fit data that can be placed on a scatter plot and create a line that “best fits” the data.

Equation: \(\text{Y} = \beta_0 +\beta_1\cdot \text{X} + \varepsilon\)

We will now illustrate an example of linear regression.
Listed below is some of the data set we will utilize for our linear regression.

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

Here is the breakdown of our linear model

Call:
lm(formula = speed ~ dist, data = cars)

Residuals:
    Min      1Q  Median      3Q     Max 
-7.5293 -2.1550  0.3615  2.4377  6.4179 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  8.28391    0.87438   9.474 1.44e-12 ***
dist         0.16557    0.01749   9.464 1.49e-12 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3.156 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

Here we illustrate the scatter plot of the speed and distance of the dataset.

ggplot(cars, aes(speed, dist)) + geom_point()

Here we have the same scatter plot with the best fit line plotted with it.

ggplot(cars, aes(speed, dist)) + geom_point() + geom_smooth(method=lm)

Equation: \(\text{dist} = \beta_0 +\beta_1\cdot \text{speed} + \varepsilon\) = 0

Equation: \(\text{uptake} = \beta_0 +\beta_1\cdot \text{conc} + \varepsilon\) = 0

data("CO2")
comod = lm(conc ~ uptake, data=CO2)
X = CO2$uptake
Y= CO2$conc
post = plot_ly(x=X, y=Y, type="scatter", mode="markers", 
        name="data", width=700, height= 400) %>%
        add_lines(x=X, y=fitted(comod), name="bestfit") %>%
        layout(margin=list(
                          l=175,
                          r=60,
                          b=30,
                          t=40
        )
        )

post