What is Simple Linear Regression?

• A statistical method for modeling the linear relationship between two variables, X and Y
• Independent Variable \(X\): eruption duration (minutes)
• Dependent Variable \(Y\): waiting time until the next eruption (minutes)
• Goal: find the line of best fit for the dataset
• Applied here to Old Faithful geyser eruption data

The Model

The simple linear regression model is defined as:

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

Where:

• \(Y\) is the dependent variable (waiting time, in minutes)
• \(X\) is the independent variable (eruption duration,in minutes)
• \(\beta_0\) is the intercept
• \(\beta_1\) is the slope
• \(\epsilon\) is the error term, assumed \(\epsilon \sim N(0, \sigma^2)\)

Estimating the Coefficients

The coefficients are estimated using the least squares method, which minimizes the sum of squared residuals.

\[\hat{\beta}_1 = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sum(x_i - \bar{x})^2}\]

\[\hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}\]

• \(\bar{x}\) and \(\bar{y}\) are the sample means
• \(\hat{\beta}_1\) is the slope — change in waiting time per minute of eruption
• \(\hat{\beta}_0\) is the intercept — expected waiting time when eruption duration is zero

Faithful Dataset

The Faithful dataset records waiting time between eruptions and eruption duration for the Old Faithful geyser in Yellowstone National Park, Wyoming, USA. It contains two variables, eruptions and waiting, both measured in minutes, with 272 observations total.

##    eruptions        waiting    
##  Min.   :1.600   Min.   :43.0  
##  1st Qu.:2.163   1st Qu.:58.0  
##  Median :4.000   Median :76.0  
##  Mean   :3.488   Mean   :70.9  
##  3rd Qu.:4.454   3rd Qu.:82.0  
##  Max.   :5.100   Max.   :96.0

Visualizing the Data

Fitting the Model

model <- lm(formula = waiting ~ eruptions, data = faithful)

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

3D Plotly plot

Interpreting the Results

• The intercept is 33.47 minutes when eruption duration is 0.it is part of the model but not physically meaningful.
• The slope of 10.73 means that for every additional minute of eruption duration waiting time increases by approximately 10.73 minutes.
• An R-squared of 0.8115 means that 81% of the variation in waiting time is explained by eruption duration indicating a strong fit.
• Eruption duration is a strong predictor of waiting time at Old Faithful.