• Statistical method that models two variables:
  • Predictor (x): the input/independent variable
  • Response (y): the output/dependent variable
• The goal is to find a line relation that best describes how X influences Y.

• Does movie length predict ratings?
• Does study time predict exam scores?

\[Y = \beta_0 + \beta_1 X + \epsilon\]

• \(Y\) is the response variable

• \(X\) is the predictor variable

• \(\beta_0\) is the intercept

• \(\beta_1\) is the slope

• \(\epsilon\) is the error term

• After fitting a regression line we want to know how good is this line actually?
• So, we use R²:

\[R^2 = 1 - \frac{SSE}{SST} = 1 - \frac{\sum_{i=1}^{n}(y_i - \hat{y}_i)^2}{\sum_{i=1}^{n}(y_i - \bar{y})^2}\]

Where:

• \(SST\) = total variance in Y

• \(SSE\) = variance unexplained by the model

• \(SST - SSE\) = variance explained by the model

movies_clean <- movies[movies$length >= 60 & movies$length <= 180, ]
model_movies <- lm(rating ~ length, data = movies_clean)
summary(model_movies)

## 
## Call:
## lm(formula = rating ~ length, data = movies_clean)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.7420 -0.9187  0.1402  1.0224  4.6929 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 4.1838120  0.0377631  110.79   <2e-16 ***
## length      0.0170546  0.0003895   43.79   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.492 on 47578 degrees of freedom
## Multiple R-squared:  0.03874,    Adjusted R-squared:  0.03872 
## F-statistic:  1917 on 1 and 47578 DF,  p-value: < 2.2e-16