library(ggplot2)
library(plotly)

This presentation covers key statistical concepts using R.

Linear regression is used to model the relationship between a dependent variable and an independent variable.

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

The least squares estimation method minimizes the sum of squared residuals:

\[ \hat{\beta_1} = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2} \]

\[ \hat{\beta_0} = \bar{Y} - \hat{\beta_1} \bar{X} \]

library(ggplot2)
data(mtcars)
head(mtcars)

##                    mpg cyl disp  hp drat    wt  qsec vs am gear carb
## Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
## Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
## Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
## Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
## Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
## Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

ggplot(mtcars, aes(x = hp, y = mpg, color = factor(cyl))) + 
  geom_point() + 
  geom_smooth(method = "lm", se = FALSE, color = "red") +
  labs(title = "Linear Regression: MPG Vs Horsepower", 
       x = "Horsepower", 
       y = "Miles Per Gallon (MPG)", 
       color = "Cylinders") + 
  theme_minimal() +
  theme(plot.title = element_text(hjust = 0.5))

ggplot(mtcars, aes(x = disp, y = mpg, color = factor(cyl))) + 
  geom_point() + 
  geom_smooth(method = "lm", se = FALSE, color = "blue") +
  labs(title = "Linear Regression: MPG vs Displacement", 
       x = "Displacement (cu. in.)", 
       y = "Miles Per Gallon (MPG)",
       color = "Cylinders") + 
  theme_minimal() +
  theme(plot.title = element_text(hjust = 0.5))

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

model <- lm(mpg ~ wt, data = mtcars)
summary(model)

## 
## Call:
## lm(formula = mpg ~ wt, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5432 -2.3647 -0.1252  1.4096  6.8727 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  37.2851     1.8776  19.858  < 2e-16 ***
## wt           -5.3445     0.5591  -9.559 1.29e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.046 on 30 degrees of freedom
## Multiple R-squared:  0.7528, Adjusted R-squared:  0.7446 
## F-statistic: 91.38 on 1 and 30 DF,  p-value: 1.294e-10

This presentation demonstrated statistical concepts using R Markdown with ioslides.