Final Project

PSTAT122: Design and Analysis of Experiments

Author

Winter 2026

STUDENT NAME

Ethan Chuang (A1V2819)
Ryan Chang (A096381)
Daniel Larson (A3E8Y73)

Due Date

Due Date: Monday, March 16, 2026, 11:59 PM

1 Introduction

In this experiment, we will test how having a “follower” car affects the distance a “lead” car can travel. More specifically, we are trying to test if the aerodynamic efficiency of the lead car can be improved by having a secondary car that is drafting behind it. Using the main effects of vehicle geometry and drafting, we will conduct a factorial experiment to test if the individual effects as well as the effect of their interaction have any affect on of the vehicle’s distance.

In the real world, shipping companies are starting to use “platoons,” where trucks drive very close together to save fuel. Our results could help show which geometric shape of vehicles save the most energy when driving in these groups in order to create more efficient transportation systems.

We predict that the biggest improvement will happen when a boxy car has a follower. This is because big cars usually are not very efficient for aerodynamics due to the way the air does not slide easily around them and creates a large vacuum behind it. Hopefully a following car should fill the vacuum and make it more efficient to move with another car than when it drives alone.

2 Experimental Design

In this experiment, we are measuring the distance from the total distance that the car is traveling in centimeters by using a tape measure that runs along the edge of the track. The total distance will help us determine if each of the factors have an impact on the aerodynamic efficiency of each car and factor.

The two factors we will be testing are car shape and drafting or not. The two different shapes of cars we will be testing is a car with a blocky shape and a car with a sleeker shape. The blocky shape has lots of concentrated surface area facing the front while the sleeker shape has surface area facing the front that is spread out to allow air to pass around it smoother.

We used a $2^2$ Factorial Design to test each possible combination of our factors, testing the slick car and blocky car, each with and without a following car.

Factor 1: Car Structure

Blocky Car
Sleek Car

Factor 2: Drafting?

With Drafting
Without Drafting

Both the car structure and drafting levels are fixed factors since we chose a car with a block design and sleek design and we choose to have drafting or not.

This made 4 different treatment groups, sleek car without drafting, sleek car with drafting, blocky car without drafting, and blocky car with drafting.

We measured the distance each of the cars starting from the top of the ramp to the final position of the tail end of the car using a tape measure that contained the distances in centimeters.

In order to account for bias and noise in this experiment, we used replication and randomization to conduct it. We ran each treatment 5 times, creating a total of 20 results, to reduce experimental variation in the conditions of the ramp, floor, and car and reduce any human error in releasing the car. We randomized the order of these 20 results by running a function in R, sample(). (Appendix 1) This was used in order to account for any lurking variables such as difference in conditions of the car as more tests were ran on them or differences in the track as more tests were ran on it.

The entire testing process took about 45 minutes in order to set up the track, ramp and measuring tape, run the experiment using the predetermined random order, and record all the results.

Description of factors and treatment structure.
Clearly state what you are measuring and the units. Examples: Number of words recalled (count), reaction time (seconds), taste rating (1–5 scale).
Identify which factors are fixed vs. random.
Description of design type (CRD, RCBD, factorial, etc.).
Explain how randomization, replication, and (if used) blocking were implemented.
Sample size: Provide number of observations per condition. Guideline: 5–10 per treatment for CRD, 3–5 blocks for RCBD, total feasible within 1 hour.

(You are encouraged to explore more resources for determining the sample size )

3 Data Collection

The experiment was conducted in the apartment of a group member on the hardwood floor of a kitchen. The ramp was constructed by books stacked on top of each other with a piece of cardboard leaning on it to create a slanted surface for the cars to run down. A track was constructed with cutting boards and the tape measure to make sure the cars ran in a semi-straight line.

Each test was run in order according to the predetermined randomization process with one group member releasing the car, one measuring the car distance, and the other recording the results.

Each car was placed with its tail end at the top of the tape measure on the ramp and was let go from the top of the ramp and allowed to roll as far as it could with minimal interference from the track. Once the car had stopped, the tail end of the car was measured and recorded. For the tests that used a following car, “drafting”, behind the lead car, a similar setup was used. The following car was setup very close behind, but not touching, the lead car and was released from supposed rest at the same time the lead car was released. When measuring the distance of the drafting car experiment, the tail end of the lead car was measured.

The measurements were done in the direction of the tape measure, not accounting for any other direction that the car might have gone in.

Some difficulties we faced when collecting data came in the form of human and random error. The person releasing the car tried to release the cars with zero initial force to the best of their ability. The car themselves tended to veer away from the track unpredictably at times, causing the measurements to be not fully correct since the car moved in a non-linear path. In order to account for this, we used another straightedge to align a right angle with the tape measure so that we would get the most accurate distance the car traveled in the direction of the tape measure.

Data Presentation:

Table of the data: (Appendix 2)

Condition	Distance (cm)
blocky car, no draft	161
sleek car, draft	271
sleek car, no draft	231
blocky car, no draft	178
blocky car, draft	216
sleek car, no draft	244
blocky car, no draft	197
blocky car, draft	224
sleek car, no draft	256
sleek car, draft	266
blocky car, draft	202
sleek car, no draft	262
blocky car, draft	252
blocky car, no draft	201
sleek car, draft	272
sleek car, draft	298
sleek car, draft	297
blocky car, draft	234
sleek car, no draft	247
blocky car, no draft	158

Boxplot of the data: (Appendix 3)

Summary of the data: (Appendix 4)

Summary Results
Condition	Trials	Mean Time	SD
blocky car, draft	5	225.6	18.84
blocky car, no draft	5	179.0	19.84
sleek car, draft	5	280.8	15.42
sleek car, no draft	5	248.0	11.90

Procedure: Describe how and when the experiment was conducted (e.g., location, date, steps taken).
Challenges/Adjustments: Mention any difficulties or changes made during data collection (e.g., technical issues, time adjustments).
Data Presentation: Display the collected data in tables or graphs, summarizing key measures like mean and standard deviation..

4 Analysis

Exploratory Data: Start with basic statistics (mean, SD) and visualizations (e.g., boxplots) to understand the data.
Hypothesis Testing: Test your hypothesis with an appropriate statistical test (e.g., ANOVA).
Tables, Figures, & Code: Include key results (ANOVA table, post-test) and relevant R code excerpts where needed.
Use R to analyze the data.

5 Conclusions

Summarize key findings.
Comment on limitations and possible improvements.

6 References

(If needed.)

7 Appendices

1. Random Assignment

Code

set.seed(67)  
treatments <- rep(c("sleek car, no draft", "sleek car, draft ", "blocky car, no draft", "blocky car, draft"), each = 5)
sample(treatments)

 [1] "blocky car, no draft" "sleek car, no draft"  "blocky car, no draft"
 [4] "sleek car, draft "    "blocky car, no draft" "blocky car, draft"   
 [7] "blocky car, draft"    "blocky car, no draft" "sleek car, no draft" 
[10] "blocky car, draft"    "blocky car, no draft" "blocky car, draft"   
[13] "sleek car, draft "    "sleek car, no draft"  "sleek car, no draft" 
[16] "sleek car, draft "    "sleek car, no draft"  "sleek car, draft "   
[19] "sleek car, draft "    "blocky car, draft"

2. Data Presentation

Code

data1 <- data.frame(
  assignment = c("blocky car, no draft", "sleek car, draft ", "sleek car, no draft", 
                 "blocky car, no draft", "blocky car, draft", "sleek car, no draft", 
                 "blocky car, no draft", "blocky car, draft", "sleek car, no draft", 
                 "sleek car, draft ", "blocky car, draft", "sleek car, no draft", 
                 "blocky car, draft", "blocky car, no draft", "sleek car, draft ", 
                 "sleek car, draft ", "sleek car, draft ", "blocky car, draft", 
                 "sleek car, no draft", "blocky car, no draft"),
  distance = c(161, 271, 231, 178, 216, 244, 197, 224, 256, 266, 202, 262, 252, 201, 272, 298, 297, 234, 247, 158)
)
knitr::kable(data1, col.names = c("Condition", "Distance (cm)"), align = "lc")

data <- data.frame(
  distance=c(161, 271, 231, 178, 216, 244, 197, 224, 256, 266, 202, 262, 252, 201, 272, 298, 297, 234, 247, 158),
  draft=c(0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0),
  shape=c(0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0)
)

3. Data Boxplot

Code

library(ggplot2)

ggplot(data1, aes(x = assignment, y = distance, fill = assignment)) +
  geom_boxplot(alpha = 0.7) + 
  geom_jitter(width = 0.1, color = "black", alpha = 0.6) +
  labs(title = "Car Distance: Impact of Shape and Drafting",
       x = "Vehicle Shape and Draft",
       y = "Distance (cm)") +
  theme_minimal() +
  theme(legend.position = "none")

4. Data Summary

Code

library(dplyr)
library(knitr)

summary_table <- data1 %>%
  mutate(assignment = trimws(assignment)) %>%
  group_by(assignment) %>%
  summarise(
    n = n(), 
    mean_time = mean(distance), 
    sd_time = sd(distance)
  )

kable(summary_table, 
      digits = 2, 
      col.names = c("Condition", "Trials", "Mean Time", "SD"),
      caption = "Summary Results")

--- title: 'Final Project' subtitle: "PSTAT122: Design and Analysis of Experiments" author: - name: "Winter 2026" # affiliations: # - name: "Fall 2025" affiliation-title: "Quarter" format: html: code-fold: true code-line-numbers: true code-copy: true code-tools: true self-contained: true toc: false toc-location: left number-sections: true geometry: margin=0.2in --- ```{r setup, include=FALSE} knitr::opts_chunk$set(message = FALSE) knitr::opts_chunk$set(warning = FALSE) knitr::opts_chunk$set(error = FALSE) ``` :::callout <div style="text-align: center"> <span style="color: blue;"> **STUDENT NAME **</span> </div> - Ethan Chuang (A1V2819) - Ryan Chang (A096381) - Daniel Larson (A3E8Y73) ::: ::: {.callout-caution} # Due Date **Due Date:** Monday, March 16, 2026, 11:59 PM ::: # Introduction In this experiment, we will test how having a "follower" car affects the distance a "lead" car can travel. More specifically, we are trying to test if the aerodynamic efficiency of the lead car can be improved by having a secondary car that is drafting behind it. Using the main effects of vehicle geometry and drafting, we will conduct a factorial experiment to test if the individual effects as well as the effect of their interaction have any affect on of the vehicle's distance. In the real world, shipping companies are starting to use "platoons," where trucks drive very close together to save fuel. Our results could help show which geometric shape of vehicles save the most energy when driving in these groups in order to create more efficient transportation systems. We predict that the biggest improvement will happen when a boxy car has a follower. This is because big cars usually are not very efficient for aerodynamics due to the way the air does not slide easily around them and creates a large vacuum behind it. Hopefully a following car should fill the vacuum and make it more efficient to move with another car than when it drives alone. # Experimental Design In this experiment, we are measuring the distance from the total distance that the car is traveling in centimeters by using a tape measure that runs along the edge of the track. The total distance will help us determine if each of the factors have an impact on the aerodynamic efficiency of each car and factor. The two factors we will be testing are car shape and drafting or not. The two different shapes of cars we will be testing is a car with a blocky shape and a car with a sleeker shape. The blocky shape has lots of concentrated surface area facing the front while the sleeker shape has surface area facing the front that is spread out to allow air to pass around it smoother. We used a $2^2$ Factorial Design to test each possible combination of our factors, testing the slick car and blocky car, each with and without a following car. **Factor 1: Car Structure** - Blocky Car - Sleek Car **Factor 2: Drafting?** - With Drafting - Without Drafting Both the car structure and drafting levels are fixed factors since we chose a car with a block design and sleek design and we choose to have drafting or not. This made 4 different treatment groups, sleek car without drafting, sleek car with drafting, blocky car without drafting, and blocky car with drafting. We measured the distance each of the cars starting from the top of the ramp to the final position of the tail end of the car using a tape measure that contained the distances in centimeters. In order to account for bias and noise in this experiment, we used replication and randomization to conduct it. We ran each treatment 5 times, creating a total of 20 results, to reduce experimental variation in the conditions of the ramp, floor, and car and reduce any human error in releasing the car. We randomized the order of these 20 results by running a function in R, sample(). (Appendix 1) This was used in order to account for any lurking variables such as difference in conditions of the car as more tests were ran on them or differences in the track as more tests were ran on it. The entire testing process took about 45 minutes in order to set up the track, ramp and measuring tape, run the experiment using the predetermined random order, and record all the results. - Description of factors and treatment structure. - Clearly state what you are measuring and the units. Examples: Number of words recalled (count), reaction time (seconds), taste rating (1–5 scale). - Identify which factors are fixed vs. random. - Description of design type (CRD, RCBD, factorial, etc.). - Explain how randomization, replication, and (if used) blocking were implemented. - Sample size: Provide number of observations per condition. Guideline: 5–10 per treatment for CRD, 3–5 blocks for RCBD, total feasible within 1 hour. (You are encouraged to explore more resources for determining the sample size ) # Data Collection The experiment was conducted in the apartment of a group member on the hardwood floor of a kitchen. The ramp was constructed by books stacked on top of each other with a piece of cardboard leaning on it to create a slanted surface for the cars to run down. A track was constructed with cutting boards and the tape measure to make sure the cars ran in a semi-straight line. Each test was run in order according to the predetermined randomization process with one group member releasing the car, one measuring the car distance, and the other recording the results. Each car was placed with its tail end at the top of the tape measure on the ramp and was let go from the top of the ramp and allowed to roll as far as it could with minimal interference from the track. Once the car had stopped, the tail end of the car was measured and recorded. For the tests that used a following car, "drafting", behind the lead car, a similar setup was used. The following car was setup very close behind, but not touching, the lead car and was released from supposed rest at the same time the lead car was released. When measuring the distance of the drafting car experiment, the tail end of the lead car was measured. The measurements were done in the direction of the tape measure, not accounting for any other direction that the car might have gone in. Some difficulties we faced when collecting data came in the form of human and random error. The person releasing the car tried to release the cars with zero initial force to the best of their ability. The car themselves tended to veer away from the track unpredictably at times, causing the measurements to be not fully correct since the car moved in a non-linear path. In order to account for this, we used another straightedge to align a right angle with the tape measure so that we would get the most accurate distance the car traveled in the direction of the tape measure. **Data Presentation:** **Table of the data:** (Appendix 2) ```{r, echo = FALSE, fig.align="center"} data1 <- data.frame( assignment = c("blocky car, no draft", "sleek car, draft ", "sleek car, no draft", "blocky car, no draft", "blocky car, draft", "sleek car, no draft", "blocky car, no draft", "blocky car, draft", "sleek car, no draft", "sleek car, draft ", "blocky car, draft", "sleek car, no draft", "blocky car, draft", "blocky car, no draft", "sleek car, draft ", "sleek car, draft ", "sleek car, draft ", "blocky car, draft", "sleek car, no draft", "blocky car, no draft"), distance = c(161, 271, 231, 178, 216, 244, 197, 224, 256, 266, 202, 262, 252, 201, 272, 298, 297, 234, 247, 158) ) knitr::kable(data1, col.names = c("Condition", "Distance (cm)"), align = "lc") data <- data.frame( distance=c(161, 271, 231, 178, 216, 244, 197, 224, 256, 266, 202, 262, 252, 201, 272, 298, 297, 234, 247, 158), draft=c(0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0), shape=c(0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0) ) ``` **Boxplot of the data:** (Appendix 3) ```{r, echo = FALSE,fig.align="center" } library(ggplot2) ggplot(data1, aes(x = assignment, y = distance, fill = assignment)) + geom_boxplot(alpha = 0.7) + geom_jitter(width = 0.1, color = "black", alpha = 0.6) + labs(title = "Car Distance: Impact of Shape and Drafting", x = "Vehicle Shape and Draft", y = "Distance (cm)") + theme_minimal() + theme(legend.position = "none") ``` \newpage **Summary of the data:** (Appendix 4) ```{r, echo = FALSE} library(dplyr) library(knitr) summary_table <- data1 %>% group_by(assignment) %>% summarise( n = n(), mean_time = mean(distance), sd_time = sd(distance) ) kable(summary_table, digits = 2, col.names = c("Condition", "Trials", "Mean Time", "SD"), caption = "Summary Results") ``` - **Procedure**: Describe how and when the experiment was conducted (e.g., location, date, steps taken). - **Challenges/Adjustments**: Mention any difficulties or changes made during data collection (e.g., technical issues, time adjustments). - **Data Presentation**: Display the collected data in tables or graphs, summarizing key measures like mean and standard deviation.. # Analysis ```{r, echo = FALSE} library(ggplot2) plot_data <- data %>% group_by(draft, shape) %>% summarize(mean_distance = mean(distance), .groups = 'drop') ggplot(plot_data, aes(x = shape, y = mean_distance, color = draft, group = draft)) + geom_line() + geom_point() + labs(title = "Interaction plot of Shape and Draft", x = "Shape (Blocky or Sleek)", y = "Mean Distance") ``` - **Exploratory Data**: Start with basic statistics (mean, SD) and visualizations (e.g., boxplots) to understand the data. - **Hypothesis Testing**: Test your hypothesis with an appropriate statistical test (e.g., ANOVA). - **Tables, Figures, & Code**: Include key results (ANOVA table, post-test) and relevant `R` code excerpts where needed. - Use `R` to analyze the data. # Conclusions - Summarize key findings. - Comment on limitations and possible improvements. # References (If needed.) # Appendices **1. Random Assignment** ```{r} set.seed(67) treatments <- rep(c("sleek car, no draft", "sleek car, draft ", "blocky car, no draft", "blocky car, draft"), each = 5) sample(treatments) ``` **2. Data Presentation** ```{r, eval = FALSE} data1 <- data.frame( assignment = c("blocky car, no draft", "sleek car, draft ", "sleek car, no draft", "blocky car, no draft", "blocky car, draft", "sleek car, no draft", "blocky car, no draft", "blocky car, draft", "sleek car, no draft", "sleek car, draft ", "blocky car, draft", "sleek car, no draft", "blocky car, draft", "blocky car, no draft", "sleek car, draft ", "sleek car, draft ", "sleek car, draft ", "blocky car, draft", "sleek car, no draft", "blocky car, no draft"), distance = c(161, 271, 231, 178, 216, 244, 197, 224, 256, 266, 202, 262, 252, 201, 272, 298, 297, 234, 247, 158) ) knitr::kable(data1, col.names = c("Condition", "Distance (cm)"), align = "lc") data <- data.frame( distance=c(161, 271, 231, 178, 216, 244, 197, 224, 256, 266, 202, 262, 252, 201, 272, 298, 297, 234, 247, 158), draft=c(0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0), shape=c(0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0) ) ``` **3. Data Boxplot** ```{r,fig.align="center", eval = FALSE} library(ggplot2) ggplot(data1, aes(x = assignment, y = distance, fill = assignment)) + geom_boxplot(alpha = 0.7) + geom_jitter(width = 0.1, color = "black", alpha = 0.6) + labs(title = "Car Distance: Impact of Shape and Drafting", x = "Vehicle Shape and Draft", y = "Distance (cm)") + theme_minimal() + theme(legend.position = "none") ``` **4. Data Summary** ```{r, eval = FALSE} library(dplyr) library(knitr) summary_table <- data1 %>% mutate(assignment = trimws(assignment)) %>% group_by(assignment) %>% summarise( n = n(), mean_time = mean(distance), sd_time = sd(distance) ) kable(summary_table, digits = 2, col.names = c("Condition", "Trials", "Mean Time", "SD"), caption = "Summary Results") ```