Simple linear regression is a statistical method that allows us to summarize and study relationships between two continuous (quantitative) variables.

The equation of a simple linear regression line is:

\[ y = \beta_0 + \beta_1 x + \epsilon \]

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

We will use a dataset that includes heights and weights.

## 
## Call:
## lm(formula = weights ~ heights, data = sample_data)
## 
## Residuals:
##       1       2       3       4       5 
## -1.5319  2.8485 -0.9383 -4.0104  3.6322 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept) -10.5983    21.2310  -0.499   0.6520  
## heights       0.4525     0.1212   3.732   0.0335 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.68 on 3 degrees of freedom
## Multiple R-squared:  0.8228, Adjusted R-squared:  0.7637 
## F-statistic: 13.93 on 1 and 3 DF,  p-value: 0.03353

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

The linear regression model can be mathematically represented as:

\[ \hat{y} = \hat{\beta_0} + \hat{\beta_1} x \]

where \(\hat{y}\) is the predicted value of \(y\), \(\hat{\beta_0}\) and \(\hat{\beta_1}\) are the estimated coefficients.

## 
## Call:
## lm(formula = weights ~ heights, data = sample_data)
## 
## Residuals:
##       1       2       3       4       5 
## -1.5319  2.8485 -0.9383 -4.0104  3.6322 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept) -10.5983    21.2310  -0.499   0.6520  
## heights       0.4525     0.1212   3.732   0.0335 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.68 on 3 degrees of freedom
## Multiple R-squared:  0.8228, Adjusted R-squared:  0.7637 
## F-statistic: 13.93 on 1 and 3 DF,  p-value: 0.03353