HW 01: Education & median income in US Counties

due Wednesday, September 21 at 11:59pm

Introduction

In this assignment, you’ll use simple linear regression to examine the association between between the percent of adults with a bachelor’s degree and the median household income for counties in the United States.

Learning goals

In this assignment, you will…

  • Fit, interpret, and evaluate simple linear regression models.
  • Conduct statistical inference for the population slope, \(\beta_1\)
  • Create and interpret spatial data visualizations using R.
  • Continue developing a workflow for reproducible data analysis.

Getting started

Log in to RStudio

  • Go to https://cmgr.oit.duke.edu/containers and login with your Duke NetID and Password.

  • Click STA210 to log into the Docker container. You should now see the RStudio environment.

Clone the repo & start new RStudio project

  • Go to the course organization at github.com/sta210-fa22 organization on GitHub. Click on the repo with the prefix hw-01. It contains the starter documents you need to complete the lab.

  • Click on the green CODE button, select Use SSH (this might already be selected by default, and if it is, you’ll see the text Clone with SSH). Click on the clipboard icon to copy the repo URL.

  • In RStudio, go to File ➛ New Project ➛Version Control ➛ Git.

  • Copy and paste the URL of your assignment repo into the dialog box Repository URL. Again, please make sure to have SSH highlighted under Clone when you copy the address.

  • Click Create Project, and the files from your GitHub repo will be displayed in the Files pane in RStudio.

  • Click hw-01.qmd to open the template R Markdown file. This is where you will write up your code and narrative for the lab.

Packages

The following packages will be used in this assignment:

library(tidyverse) # for data wrangling
library(tidymodels) # for modeling and inference
library(knitr) # to neatly format tables
library(scales) # to format visualizations

Data: US Counties

The data are from the county_2019 data frame in theusdata R package. These data were originally collected in the 2019 American Community Survey (ACS), an annual survey conducted by the United States Census Bureau that collections demographics and other information from a sample of households in the United States. The data in county_2019 are county-level statistics from the ACS.

The data you will use in this analysis are available in the file us-counties-sample.csv; it contains a random sample of 600 counties in the United States.

The primary variables you will use are the following:

  • bachelors: Percent of population 25 and older that earned a Bachelor’s degree or higher
  • median_household_income: Median household income
  • household_has_computer: Percent of households that have desktop or laptop computer

Click here for the full codebook for the county_2019 dataset.

You will use two other data sets county-map-sample.csv and county-map-all.csv to help create spatial visualizations of the ACS variables. Use the code below to load all of the data sets.

county_data_sample <- read_csv("data/us-counties-sample.csv")
map_data_sample <-  read_csv("data/county-map-sample.csv")
map_data_all <- read_csv("data/county-map-all.csv")

Exercises

There has been a lot of conversation recently about the impact of graduating college, i.e., obtaining a bachelor’s degree, on one’s future career and lifetime earnings. The common convention is that individuals who have earned a bachelor’s degree (or higher) will earn more income over the course of a lifetime than an individual who does not have such a degree.

While we don’t have individual data in this analysis, we will examine the association between these two factors at a county level. In other words, do counties that have a higher percentage of college graduates have higher median household incomes, on average, compared to counties with a lower percentage of college graduates?

Important

All narrative should be written in complete sentences, and all visualizations should have informative titles and axis labels.

Part 1: Exploratory data analysis

Exercise 1

Create a histogram of the distribution of the response variable median_household_income and calculate appropriate summary statistics. Use the visualization and summary statistics to describe the distribution. Include an informative title and axis labels on the plot.

Exercise 2

Let’s view the data in another way. Use the code below to make a map of the United States with the color of the counties filled in based on the median household income. Fill in title and axis labels.

Then use the plot answer the following:

  • What are 2 - 3 observations you have from the map?
  • What is a feature that is apparent in the map that wasn’t as easily apparent from the histogram in the previous exercise? What is a feature that is apparent in the histogram that is not as easily apparent from the map?
county_map_data <- left_join(county_data_sample, map_data_sample)

ggplot(data = map_data_all) +
  geom_polygon(aes(x = long, y = lat, group = group),
    fill = "lightgray", color = "white"
    ) +
  geom_polygon(data = county_map_data, aes(x = long, y = lat, group = group,
    fill = median_household_income)
    ) +
  labs(
    x = "___",
    y = "___",
    fill = "___",
    title = "___"
  ) +
  scale_fill_viridis_c(labels = label_dollar()) +
  coord_quickmap()

Exercise 3

Create a visualization of the relationship between bachelors and median_household_income . Use the visualization to describe the relationship between the two variables.

If you haven’t yet done so, now is a good time to render your document and commit (with a meaningful commit message) and push all updates.

Part 2: Modeling

Exercise 4

We will use a linear regression model to better quantify the relationship between bachelors and median_household_income.

Write the form of the statistical model we will use for this task using mathematical notation. Use variable names (bachelors and median_household_income) in the equation for your model1.

Exercise 5

  • Fit the linear model to understand variability in the median household income based on the percent of adults age 25 and older in the county with a bachelor’s degree. Neatly display the model output with 3 digits.

  • Write the estimated regression equation using mathematical notation. Use variable names (bachelors and median_household_income) in the equation.

Exercise 6

Now let’s use the model coefficients to describe the relationship between these two variables.

  • Interpret the slope. The interpretation should be written in a way that is meaningful in the context of the data.
  • Is it meaningful to interpret the intercept for this data? If so, write the interpretation in the context of the data. Otherwise, briefly explain why not.

Now is a good time to render your document again if you haven’t done so recently and commit (with a meaningful commit message) and push all updates.

Part 3: Inference for the U.S.

We want to use the data from these 600 randomly selected counties to draw conclusions about the relationship between the percent of adults age 25 and older with a bachelor’s degree and median household income for the over 3,000 counties in the United States.

Exercise 7

  • What is the population of interest? What is the sample?

  • Is it reasonable to treat the sample in this analysis as representative of the population? Briefly explain why or why not.

Exercise 8

Conduct a hypothesis test for the slope to assess whether there is sufficient evidence of a linear relationship between the percent of adults age 25 and older with a bachelor’s degree and the median household income in a county. Use a randomization (permutation) test. In your response:

  • State the null and alternative hypotheses in words and mathematical notation
  • Show all relevant code and output used to conduct the test. Use set.seed(8).
  • State the conclusion in the context of the data.

Exercise 9

Next, construct a 95% confidence interval for the slope using bootstrapping with set.seed(9). Show all relevant code and output used to calculate the interval. Interpret the confidence interval in the context of the data.

Exercise 10

Comment on whether the hypothesis test and confidence interval support the general consensus that adults who have a completed a bachelor’s degree generally earn higher income than adults who have not. A brief explanation is sufficient but it should be based on your conclusions from the hypothesis test and confidence interval.

Now is a good time to render your document again if you haven’t done so recently and commit (with a meaningful commit message) and push all updates.

Part 4: Model comparison

Exercise 11

A researcher suggests that knowing the percentage of households in a county with a computer is a better indicator of median household income than the percentage of adults with a bachelor’s degree.

  • Fit a linear model of the relationship between median_household_income and household_has_computer. Neatly display the model with 3 digits.

  • Evaluate this model and the model fit in Exercise 5 to assess the researcher’s claim.

  • Do your analysis results support the researcher’s claim? Briefly explain why or why not.

Before submitting, make sure you render your document and commit (with a meaningful commit message) and push all updates.

Submission

Warning

Before you wrap up the assignment, make sure all documents are updated on your GitHub repo. We will be checking these to make sure you have been practicing how to commit and push changes.

Remember – you must turn in a PDF file to the Gradescope page before the submission deadline for full credit.

To submit your assignment:

  • Go to http://www.gradescope.com and click Log in in the top right corner.
  • Click School Credentials ➡️ Duke NetID and log in using your NetID credentials.
  • Click on your STA 210 course.
  • Click on the assignment, and you’ll be prompted to submit it.
  • Mark the pages associated with each exercise. All of the pages of your lab should be associated with at least one question (i.e., should be “checked”).
  • Select the first page of your PDF submission to be associated with the “Workflow & formatting” section.

Grading (50 points)

Component Points
Ex 1 - 11 47
Workflow & formatting 32

Footnotes

  1. Click here for a guide on writing mathematical symbols using LaTex.↩︎

  2. The “Workflow & formatting” grade is to assess the reproducible workflow. This includes having at least 3 informative commit messages and updating the name and date in the YAML.↩︎