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1 1. Introduction

This report demonstrates the standard format for STA 506 assignments using R Markdown. The purpose of this document is to show how narrative text, statistical analysis, tables, and figures can be combined into a professional report.

We analyze a built-in dataset to illustrate the workflow.

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2 2. Data Description

For this example, we use the built-in mtcars dataset, which contains information about fuel efficiency and design characteristics of automobiles.

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

The dataset contains 32 observations and 11 variables.

Key variables include:

• mpg: Miles per gallon
  
• hp: Horsepower
  
• wt: Weight (1000 lbs)
  
• cyl: Number of cylinders

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3 3. Exploratory Data Analysis

3.1 3.1 Summary Statistics

We begin by examining summary statistics.

summary(mtcars[, c("mpg", "hp", "wt")])

      mpg              hp              wt       
 Min.   :10.40   Min.   : 52.0   Min.   :1.513  
 1st Qu.:15.43   1st Qu.: 96.5   1st Qu.:2.581  
 Median :19.20   Median :123.0   Median :3.325  
 Mean   :20.09   Mean   :146.7   Mean   :3.217  
 3rd Qu.:22.80   3rd Qu.:180.0   3rd Qu.:3.610  
 Max.   :33.90   Max.   :335.0   Max.   :5.424  

We observe that fuel efficiency varies substantially across vehicles, with horsepower and weight showing wide ranges.

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3.2 3.2 Correlation Analysis

Next, we examine correlations among key variables.

cor(mtcars[, c("mpg", "hp", "wt")])

           mpg         hp         wt
mpg  1.0000000 -0.7761684 -0.8676594
hp  -0.7761684  1.0000000  0.6587479
wt  -0.8676594  0.6587479  1.0000000

There appears to be a strong negative relationship between fuel efficiency and both horsepower and vehicle weight.

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3.3 3.3 Data Visualization

We visualize these relationships using scatterplots.

plot(
  mtcars$wt,
  mtcars$mpg,
  xlab = "Weight (1000 lbs)",
  ylab = "Miles Per Gallon",
  main = "MPG vs Vehicle Weight",
  pch = 19
)
abline(lm(mpg ~ wt, data = mtcars), col = "red")

Fuel Efficiency vs Weight

The plot indicates that heavier vehicles tend to have lower fuel efficiency.

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4 4. Statistical Modeling

4.1 4.1 Linear Regression Model

We fit a linear regression model to predict fuel efficiency.

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

Call:
lm(formula = mpg ~ wt + hp, data = mtcars)

Residuals:
   Min     1Q Median     3Q    Max 
-3.941 -1.600 -0.182  1.050  5.854 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 37.22727    1.59879  23.285  < 2e-16 ***
wt          -3.87783    0.63273  -6.129 1.12e-06 ***
hp          -0.03177    0.00903  -3.519  0.00145 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.593 on 29 degrees of freedom
Multiple R-squared:  0.8268,    Adjusted R-squared:  0.8148 
F-statistic: 69.21 on 2 and 29 DF,  p-value: 9.109e-12

The model relates miles per gallon to vehicle weight and horsepower.

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4.2 4.2 Model Interpretation

Based on the estimated coefficients:

• Weight has a strong negative effect on fuel efficiency
  
• Horsepower also contributes negatively
  
• Both predictors are statistically significant

This suggests that heavier and more powerful vehicles consume more fuel.

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5 5. Model Diagnostics

We assess model assumptions using diagnostic plots.

par(mfrow = c(2,2))
plot(model)

par(mfrow = c(1,1))

Residual plots indicate reasonable linearity and homoscedasticity.

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6 6. Results Summary

Our analysis shows that:

1. Vehicle weight is the strongest predictor of MPG
   
2. Horsepower provides additional explanatory power
   
3. The fitted model explains a substantial portion of variation

These results are consistent with physical expectations.

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7 7. Conclusion

This report demonstrates the standard structure for STA 506 assignments.

Using R Markdown allows for:

• Reproducible analysis
  
• Integrated narrative and code
  
• Professional formatting

Future assignments will follow this template.

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8 Appendix (Optional)

Additional analyses and code may be placed here if needed.

mean(mtcars$mpg)

[1] 20.09062

sd(mtcars$mpg)

[1] 6.026948