• This presentation will cover the topic of Applied Regression. Applied regression is a statistical method used to analyze relationships between variables and make predictions using real world data.

Rather than focusing only on theory, applied regression helps answer practical questions such as:

• How does one variable affect another?
• Can we predict an outcome using available data?
• Which factors are most important in explaining a result?

In this presentation, we use applied regression to study how car weight affects fuel efficiency, from the mtcars dataset.

Simple linear regression model: \[Y = \beta_0 + \beta_1 X + \epsilon\] Where: - \(Y\): response variable (MPG) - \(X\): predictor variable (weight) - \(\beta_0, \beta_1\): model parameters - \(\epsilon\): error term — I use the built-in mtcars dataset in R. Variables used: - mpg = miles per gallon - wt = car weight The data is as follows:

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

There is a negative relationship between weight and MPG. Generally, heavier cars tend to have lower fuel efficiency. The pattern looks linear, making linear regression an appropriate model.

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

The estimated regression model is: \[ \hat{Y} = \hat{\beta}_0 + \hat{\beta}_1 X \] This equation represents the fitted line used to predict MPG based on car weight.

• \(\hat{\beta}_0\): estimated intercept
  
• \(\hat{\beta}_1\): estimated slope
  

## 
## 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

The interactive plot shows a negative relationship between car weight and MPG. As weight increases, fuel efficiency decreases, which corroborates the earlier scatterplot. The interactive features allow us to explore individual data, but the overall downward trend remains clear.

Applied regression provides a practical way to understand relationships in data and make actionable predictions. In this analysis, car weight is a strong predictor for fuel efficiency. The regression model showed an obvious negative or downward trend, showing that increases in weight are associated with decreases in fuel efficiency or MPG. The results show this relationship:

• The slope is negative and statistically significant
• The \(R^2\) value suggests that a most of the variability in MPG is explained by weight

This illustrated how regression can be used to identify patterns, and also to quantify relationship. It helps with making decisions about the data. While weight is an important factor, other variables (engine size or horsepower) may also change fuel efficiency. This presents an opportunity for future, more complex multiple regression models.