##    hours score
## 1      1    52
## 2      2    55
## 3      3    61
## 4      4    63
## 5      5    68
## 6      6    72
## 7      7    75
## 8      8    81
## 9      9    85
## 10    10    88

Here is the equation for a simple linear regression model: \[ \hat{y} = b_0 + b_1 x \]

where:

• \(\hat{y}\) is the predicted value

• \(b_0\) is the intercept

• \(b_1\) is the slope

• \(x\) is the predictor variable

The slope tells us how much the response variable changes for each one-unit increase in \(x\).

We can use R to compute the regression line using the lm() function.

\[\hat{y} = 47.53 + 4.08x\]

This plot shows the fitted regression line along with the data.

Residuals are the differences between the observed values and the predicted values.

\[ e_i = y_i - \hat{y}_i \]

A good model has residuals that are relatively small and randomly scattered around 0.

This interactive plot shows the data points and fitted regression line.

Simple linear regression is a useful statistical method for understanding the relationship between two variables. In this example, we saw that as hours studied increase, exam scores also increase.

This model allows us to:

• describe relationships between variables

• make predictions

• identify trends in data

Overall, simple linear regression is a widely applicable topic that can be used in variety of fields and industries.