Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables, assuming a linear relationship. It helps predict the value of a dependent variable based on the values of the independent variables. Essentially, it finds the best-fit line or plane that represents the data, allowing for predictions and understanding of how independent variables influence the dependent variable.

\[ \text {This is what the formula looks like: } y = \beta_0 + \beta_1 x + \varepsilon \]

The data set Diamond contains two columns with carat and price of the diamond

carat is the weight of the diamond price is the price of the diamond

We will find out for each influences each other using linear regression

##   carat price
## 1  0.17   355
## 2  0.16   328
## 3  0.17   350
## 4  0.18   325
## 5  0.25   642
## 6  0.16   342

Interactive plot graph where you can individually see each plot and can compare up and down the Regression Line

This shows the price of a diamond based on the weight in carrots We can see how the higher the weight the higher the price

Adding a linear regression line on top of the Scatterplot we already have

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

The slope \(\beta _{1}\) represents the average change in the dependent variable \(y\) for a one-unit increase in the independent variable \(x\). It indicates how much \(y\) is expected to change for each unit change in \(x\). google.com

Formula: \[ \beta_1 = \frac{ Cov(X, Y)}{Var(X) } \] Dividing 2 covariance of x and y by an independent variable x gives us \(\beta_1\) or slope. If greater than 0 then its positive if less then negative

Contains Residual error and had min, quatertile 1, median, 3rd quatile , and max.

## 
## Call:
## lm(formula = price ~ carat, data = diamond)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -85.159 -21.448  -0.869  18.972  79.370 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -259.63      17.32  -14.99   <2e-16 ***
## carat        3721.02      81.79   45.50   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 31.84 on 46 degrees of freedom
## Multiple R-squared:  0.9783, Adjusted R-squared:  0.9778 
## F-statistic:  2070 on 1 and 46 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = price ~ carat, data = diamond)
## 
## Coefficients:
## (Intercept)        carat  
##      -259.6       3721.0