• Simple Linear Regression is a statistical method to model the relationship between two variables: one independent variable and one dependent variable.
• Both variables should be quantitative.
• Predicts the value of unknown data by using another related and known data value

1. Linearity: The relationship between X and the mean of Y is linear
2. Independence: The variance of residual is the same for any value of X.
3. Homoscedasticity: Observations are independent of each other.
4. Normality: For any fixed value of X, Y is normally distributed.

## A marker object has been specified, but markers is not in the mode
## Adding markers to the mode...

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

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

The equation of a simple linear regression model can be written as:

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

where: - \(Y\) is the dependent variable, - \(X\) is the independent variable, - \(\beta_0\) is the intercept, - \(\beta_1\) is the slope coefficient, and - \(\varepsilon\) is the error term.

This equation represents the relationship between the independent variable \(X\) and the dependent variable \(Y\) in the context of linear regression.

The probability density function (PDF) of the normal distribution is given by:

\[ f(x|\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp \left( -\frac{(x-\mu)^2}{2\sigma^2} \right) \]

where: - \(x\) represents the random variable, - \(\mu\) is the mean of the distribution, and - \(\sigma^2\) is the variance.

This equation describes the shape of the normal distribution curve, which is symmetric and bell-shaped.

suppressMessages(library(plotly)) # Load necessary libraries

hours <- c(2, 3, 4, 5, 6, 7, 8) # Example dataset (2D)
scores <- c(56, 67, 75, 82, 88, 93, 97)

data <- data.frame(hours, scores) # Create a dataframe

model <- lm(scores ~ hours, data = data) # Fit linear regression model

predictions <- predict(model) # Predict scores using the regression model

# Create 3D scatter plot with regression line
plot_ly(data, x = ~hours, y = ~scores, type = "scatter3d", mode = "markers", 
marker = list(size = 5, color = "blue"), text = NULL) %>% 
add_trace(x = ~hours, y = ~predictions, z = ~predictions, type = "scatter3d", 
mode = "lines", line = list(color = "red", width = 2), text = NULL)