• Simple linear regression is a statistical method used to model the relationship between two quantitative variables: one independent variable (X) and one dependent variable (Y).

The simple linear regression model can be represented as:

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

• \(Y\) is the dependent variable
• \(X\) is the independent variable
• \(\beta_0\) is the intercept
• \(\beta_1\) is the slope
• \(\epsilon\) is the error term

In simple linear regression, the parameters \(\beta_0\) and \(\beta_1\) are estimated using least squares estimation. The formulas for estimating these parameters are:

\[ \hat{\beta_1} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n} (x_i - \bar{x})^2} \]

\[ \hat{\beta_0} = \bar{y} - \hat{\beta_1}\bar{x} \]

where: \(\hat{\beta_1}\) is the estimated slope. \(\hat{\beta_0}\) is the estimated intercept. \(\bar{x}\) is the mean of the independent variable \(X\). \(\bar{y}\) is the mean of the dependent variable \(Y\). \(n\) is the number of observations. \(x_i\) and \(y_i\) are the individual observations of \(X\) and \(Y\) respectively.

Once the parameters \(\beta_0\) and \(\beta_1\) are estimated, the prediction equation for simple linear regression is given by:

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

where: \(\hat{Y}\) is the predicted value of the dependent variable \(Y\). \(\hat{\beta_0}\) and \(\hat{\beta_1}\) are the estimated intercept and slope respectively. \(X\) is the value of the independent variable for which the prediction is being made.

This equation allows us to predict the value of the dependent variable \(Y\) for any given value of the independent variable \(X\) based on the estimated parameters of the regression model.

Let’s consider a hypothetical dataset of house prices ($) and their corresponding areas (sq. ft). We want to predict house prices based on the area.

• Below is the R code for creating house_data shared in previous slide.

# Example dataset
house_data <- data.frame(area = c(1200, 1400, 1600, 1800, 2000, 2200, 2400, 2600, 2800, 3000),
                         price = c(150000, 170000, 190000, 210000, 230000, 250000, 270000, 290000, 310000, 330000))

• Below is the scatter plot for the housing data.

• Let’s visualize the relationship between house prices, areas, and another variable (e.g., number of bedrooms).

## 
## Call:
## lm(formula = price ~ area, data = house_data)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -2.369e-11 -1.433e-11 -2.239e-12  1.049e-11  4.196e-11 
## 
## Coefficients:
##              Estimate Std. Error   t value Pr(>|t|)    
## (Intercept) 3.000e+04  2.477e-11 1.211e+15   <2e-16 ***
## area        1.000e+02  1.138e-14 8.789e+15   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.067e-11 on 8 degrees of freedom
## Multiple R-squared:      1,  Adjusted R-squared:      1 
## F-statistic: 7.724e+31 on 1 and 8 DF,  p-value: < 2.2e-16

• Suppose we want to predict the price of a house with an area of 1500 sq. ft using our linear regression model.
• R code: predicted_price <- predict(model, newdata = data.frame(area = new_area))

##      1 
## 180000

• Simple linear regression is a powerful tool for understanding and predicting the relationship between two variables.
• It provides insights into the direction and strength of the relationship, allowing for informed decision-making in various fields.