Simple linear regression is a statistical method used to study the relationship between two variables.
The data in this presentation are example data created for demonstration.
The general simple linear regression model is:
\[ Y = \beta_0 + \beta_1X + \epsilon \]
Where:
The fitted regression line is written as:
\[ \hat{Y} = b_0 + b_1X \]
We will look at the relationship between study time and exam scores.
The data suggest that students who study more hours tend to have higher exam scores.
| Hours Studied | Exam Score |
|---|---|
| 1 | 55 |
| 2 | 58 |
| 3 | 65 |
| 4 | 70 |
| 5 | 76 |
| 6 | 82 |
| 7 | 86 |
| 8 | 90 |
| 9 | 94 |
The plot shows a positive linear relationship. As study hours increase, exam scores generally increase.
For this example, the fitted regression line is approximately:
\[ \hat{Y} = 49.79 + 4.82X \]
The slope is approximately 4.82.
This means that for each additional hour studied, the predicted exam score increases by about 4.82 points, on average.
The intercept is about 49.79, which is the predicted score when study time is zero hours.
A residual is the difference between an observed value and its predicted value:
\[ e_i = y_i - \hat{y}_i \]
A residual tells us how far an observation is from the regression line.
The residuals are centered around zero, which is useful when checking whether a linear model is reasonable.
The lm() function creates the simple linear regression model.
# Create the regression model model <- lm(score ~ hours, data = study_data) # Display the model results summary(model) # Create a scatterplot with a regression line ggplot(study_data, aes(x = hours, y = score)) + geom_point() + geom_smooth(method = "lm")
R makes it easy to create the model and visualize the relationship.
Move your mouse over the points to see the study hours and exam score.
Simple linear regression helps describe and predict the relationship between two variables.
Key points:
In this example, more study time was associated with higher predicted exam scores.