Simple Linear Regression is a statistical method used to model the relationship between two variables:

• Independent variable (X)
• Dependent variable (Y)

It assumes a linear relationship between X and Y.

The simple linear regression model is represented by:

\[Y = \beta_0 + \beta_1X + \epsilon\]

Where: - \(\beta_0\) is the y-intercept - \(\beta_1\) is the slope - \(\epsilon\) is the error term

1. Linearity: The relationship between X and Y is linear
2. Independence: Observations are independent of each other
3. Homoscedasticity: Constant variance of residuals
4. Normality: Residuals are normally distributed

Let’s use a dataset of height (X) and weight (Y) to demonstrate simple linear regression.

set.seed(123)
height <- runif(100, 150, 200)
weight <- 0.5 * height + rnorm(100, 0, 5)
data <- data.frame(height, weight)

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

summary(model)

## 
## Call:
## lm(formula = weight ~ height, data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.1899  -3.0661  -0.0987   2.9817  11.0861 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.30267    5.99902   0.217    0.829    
## height       0.49102    0.03418  14.365   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.846 on 98 degrees of freedom
## Multiple R-squared:  0.678,  Adjusted R-squared:  0.6747 
## F-statistic: 206.3 on 1 and 98 DF,  p-value: < 2.2e-16

The estimated regression equation:

\[\hat{Y} = \hat{\beta_0} + \hat{\beta_1}X\]

Where: - \(\hat{\beta_0} = -31.62\) (estimated y-intercept) - \(\hat{\beta_1} = 0.51\) (estimated slope)

Therefore, our regression equation is:

\[\hat{Y} = -31.62 + 0.51X\]

Interpretation: - For every 1 cm increase in height, we expect an average increase of 0.51 kg in weight. - The expected weight for a person with 0 cm height would be -31.62 kg (not meaningful in this context).