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• Introduction to Simple Linear Regression?
• The Mathematical Equation
• An Example Dataset
• Fitting The Model
• Visualization With ggplot
• Residual Analysis
• 3D Visualization With Plotly
• R Code
• Analysis of the Results and Conclusion

• Simple Linear Regression is a statistical method used to model the relationship between independent and dependent variables
• It assumes that there is a linear relationship between the two
• It predicts Y based on X

• For Linear Regression: \[Y_i = \beta_0 + \beta_1 X_i + \epsilon_i\]
• \(Y_i\) is the dependent variable
• \(X_i\) is the independent variable
• \(\beta_0\) is the y-intercept
• \(\beta_1\) is the slope
• \(\epsilon_i\) is the random error

We can explore the relationship between speed and distance in the cars dataset

summary(cars)

##      speed           dist       
##  Min.   : 4.0   Min.   :  2.00  
##  1st Qu.:12.0   1st Qu.: 26.00  
##  Median :15.0   Median : 36.00  
##  Mean   :15.4   Mean   : 42.98  
##  3rd Qu.:19.0   3rd Qu.: 56.00  
##  Max.   :25.0   Max.   :120.00

car_model = lm(dist ~ speed, data = cars)
summary(car_model)

## 
## Call:
## lm(formula = dist ~ speed, data = cars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -29.069  -9.525  -2.272   9.215  43.201 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -17.5791     6.7584  -2.601   0.0123 *  
## speed         3.9324     0.4155   9.464 1.49e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.38 on 48 degrees of freedom
## Multiple R-squared:  0.6511, Adjusted R-squared:  0.6438 
## F-statistic: 89.57 on 1 and 48 DF,  p-value: 1.49e-12

library(ggplot2)
ggplot(cars, aes(x = speed, y = dist)) + 
  geom_point(color = "#8C1D40", alpha = 0.7) + 
  geom_smooth(method = "lm", color = "blue") + 
  labs(title = "Car Speed VS Distance", x = "Speed in mph", y = "Stopping Distance in ft", caption = "Source: 'cars' dataset in R") + 
  theme_minimal()

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

model_data = data.frame(Speed = cars$speed, Fitted = fitted(car_model), Residuals = residuals(car_model))

ggplot(model_data, aes(x = Fitted, y = Residuals)) +
  geom_point(color = "#8C1D40") +
  geom_hline(yintercept = 0, linetype = "dashed") +
  labs(
    title = "Residuals VS. Fitted Values",
    x = "Fitted Values",
    y = "Residuals"
  ) +
  theme_minimal()

library(plotly)
cars_data = cars
cars_data$fitted = fitted(car_model)

plot_ly(cars_data, x = ~speed, y = ~dist, z = ~fitted, type = 'scatter3d', mode = 'markers', marker=list(color = '#8C1D40', size = 5)) %>%
  layout(
    scene = list(
      x_axis = list(title = 'Speed in mph'),
      y_axis = list(title = 'Actual Distance in ft'),
      z_axis = list(title = 'Fitted Distance in ft')
    ),
    title = '3D Visualization of Linear Regression'
  )

library(ggplot2)
ggplot(cars, aes(x=speed, y = dist)) +
  geom_point(color = '#8C1D40') +
  geom_smooth(method = 'lm', color = 'blue') +
  ggtitle("Scatter Plot with Regression Line") +
  xlab("Speed in mph") +
  ylab("Stopping Distance in ft")

The regression model gives us the following equation: \[Distance = -17.579 + 3.932 \times Speed\] Some things we can gather from these results: - Correlation Coefficient: The p-value for the speed coefficient is extremely small (2.2e-16) meaning that there is a statistically significant relationship between the independent and dependent variable - R-squared: The speed of 0.6511 explains 65% of the variation in stopping distance - Intercept: The negative intercept (-17.579) explains that the model predicts negative distances at low speeds, which isn’t physically possible so therefore meaningless. - Practical Interpretation: For each 1 mph increase in speed, the stopping distance increases by approximately 3.9 feet.

Therefore, in conclusion, we can see that the quality of prediction in the data depends on how well the data satisfies assumptions of linear relationships. The model performed reasonably well and showed statistical significance. However, the residuals don’t suggest a good relationship between the data in the sense of it being perfectly linear.