• Introduction to regression analysis.
• Importance of simple linear regression in statistics.

• Simple linear regression is a statistical method to model the relationship between two variables. It assumes that there is a linear relationship between the predictor variable (X) and the response variable (Y).

The simple linear regression model can be represented 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, and \(\varepsilon\) is the error term.

Let’s consider an example where we want to predict the sales of a product based on the advertising budget spent on it.

# Generate example data
set.seed(123)
budget <- seq(100, 500, by = 50)
sales <- 100 + 0.5 * budget + rnorm(length(budget), mean = 0, sd = 20)
data <- data.frame(Budget = budget, Sales = sales)

# Display the first few rows of the data
head(data)

##   Budget    Sales
## 1    100 138.7905
## 2    150 170.3965
## 3    200 231.1742
## 4    250 226.4102
## 5    300 252.5858
## 6    350 309.3013

model <- lm(Sales ~ Budget, data = data)

summary(model)

## 
## Call:
## lm(formula = Sales ~ Budget, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -23.789 -11.413  -2.626   9.344  33.040 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 110.9711    17.5711   6.316 0.000398 ***
## Budget        0.4723     0.0538   8.778 5.02e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 20.84 on 7 degrees of freedom
## Multiple R-squared:  0.9167, Adjusted R-squared:  0.9048 
## F-statistic: 77.05 on 1 and 7 DF,  p-value: 5.017e-05

• The intercept \(\beta_0\) is approximately 100. This represents the estimated sales when the advertising budget is zero.
• The slope \(\beta_0\) is approximately 0.5. This indicates that for every unit increase in the advertising budget, the sales increase by an average of 0.5 units.

plot_ly(data, x = ~Budget, y = ~Sales, type = "scatter", mode = "markers", name = "Data") %>%
  add_trace(x = budget, y = predict(model), mode = "lines", name = "Regression Line") %>%
  layout(title = "Fitted Regression Line", xaxis = list(title = "Advertising Budget"), yaxis = list(title = "Sales"))

In this presentation, we introduced the concept of simple linear regression, demonstrated its application using an example, and interpreted the results obtained from the fitted regression model.