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

This equation represents a simple linear regression model which includes the following elements:

• dependent variable Y
• independent variable X
• the constant, \(\beta_0\)
• the coefficient on X, \(\beta_1\)

\(\beta_0\) and \(\beta_1\) is estimated by \(\hat{\beta}_0\) and \(\hat{\beta}_1\) respectively using the least squares method.

The fitted or estimated line is as follows: \[ y_i = \hat{\beta}_0 + \hat{\beta}_1 x_i + e_i \] (\(x_i\), \(y_i\)) is the values of data point/observation.

\(e_i\) is the residual for observation i ( \(y_i\) - \(\hat{y}_i\)).

\(\hat{y}_i\) is the estimate. \[ \hat{y}_i = \hat{\beta}_0 + \hat{\beta}_1 x_i \]

x = cars$speed
y = cars$dist
df = data.frame(x = x, y = y)
lm_model = lm(y ~ x, data = df)
intercept = coef(lm_model)[1]
slope = coef(lm_model)[2]
plot = ggplot(df, aes(x = x, y = y)) +
  geom_point() +  # Add points
  geom_smooth(method = "lm", se = FALSE, formula = y ~ x) +
  geom_text(aes(x = max(x), y = max(y), 
                label = paste("y =", round(intercept, 2),
                              "+", round(slope, 2), "* x")),
            
            hjust = 1, vjust = 1)

\(\hat{y}_i\) = -17.58 + 3.93x

This is a possible hypothesis test to determine if there is or if there is not evidence that X and Y have a linear relationship. We can use correlation, (\(\rho\)), or \(\beta_1\) as means for the null and alternative hypothesis.

\(\rho\) is a measure of how strong or weak the linear relationship is between X and Y. \[ -1 \leq \rho \leq 1 \]

\(R^2\) is the fraction of variability that can be accounted for by the regression model.

\[ 0 \leq R^2 \leq 1 \] We want \(R^2\) to be high. An \(R^2\) value > 0.7 indicates a strong correlation. Let’s use our car data:

## [1] 0.6510794

\(R^2\) is fairly close to 0.7 indicating a strong correlation.

We want the residuals \(e_i\) to be:

• normally distributed
• have constant variance
• have a mean equal to 0.

Mean of Residuals of Previous Car data:

## [1] -4.440892e-16

It is very close to 0 so it is a good indicator. Let’s check the other two conditions using the same car data from before.

It is somewhat skewed but relatively follows the normal distribution.

Yes, the data looks scattered and does not seem to follow a pattern.

• Scatterplot appears to show a linear relationship between car speed and distance.
• \(R^2\) is high showing that the linear relationship is strong.
• Residuals appear to be normally distributed, have constant variance, and have a mean close to 0.

Inference: A simple linear regression model seems to be an excellent model of car speed as a function of car distance.