Sta-209 (Fall 2025) Homework #9

Directions:

• Submit your work via the “Assignments” tab on Canvas
• For this assignment you should record your answers/code using R Markdown
  • Please upload HTML, Word, or PDF output created using R Markdown and make sure it contains your code, output, and written answers. You should not include extraneous output, such as printing an entire data frame.
  • At this point in the course you are responsible for knowing how to properly knit an R Markdown document, so uploading any other file format will result in a point deduction on the assignment
• Homework is an individual assignment. It’s okay to check your work or collaborate with your classmates, student mentors, and others, but it is not okay to pass off their work as your own.
  • Please clearly acknowledge any help you get from individuals other than yourself, or resources other than the materials on our course website (such as external websites and AI)

Question #1

The Armed Forces Qualification Test (AFQT) is a multiple-choice test designed to measure an individual’s reasoning and verbal skills to assess their suitability for various military service roles. Throughout the 1970s the test was applied very broadly, which led to percentile scores on it becoming a commonly used measure of general intelligence.

The data provided below are a random sample of \(n=2584\) Americans who took the AFQT in 1979. They were later interviewed in 2006 about their educational attainment and annual income.

afqt <- read.csv("https://remiller1450.github.io/data/AFQT.csv")

• Part A: Create histogram of the outcome variable Income2005. Briefly describe the shape of this distribution.
• Part B: Create a scatter plot showing the relationship between AFQT (x-axis), Income2005 (y-axis), and number of years of education completed (the color aesthetic). Based upon this graph, which combinations of variables appear related? You should clearly indicate whether you see an association between all three combinations (AFQT and Income2005, AFQT and Educ, Educ and Income2005) and briefly describe the direction of that association (ie: higher AFQT is associated with higher income, etc.)
• Part C: Fit a simple linear regression model using AFQT to predict Income2005. Report and interpret the marginal effect (slope coefficient) from the fitted model.
• Part D: Now fit a multivariable linear regression model that uses both AFQT and Educ to predict Income2005. Report and interpret the adjusted effect of AFQT after adjusting for differences in educational attainment.
• Part E: Comparing Part C and Part D, you should notice that the marginal effect of a 1-percentile increase in AFQT is roughly twice as large as the adjusted effect of a 1-percentile increase in AFQT. In 1-2 sentences, explain why this is. That is, what is it about the relationships in these data that would lead to this type of difference?
• Part F: Use an \(F\)-test to compare the models Income2005 ~ AFQT + Educ and Income2005 ~ AFQT. Report the \(p\)-value of this test and provide a brief conclusion.
• Part G: Check the assumptions needed for the test you performed in Part F to produce an accurate \(p\)-value. You should clearly state each assumption and justify whether or not you believe it to be reasonable by referencing an appropriate diagnostic plot.
• Part H: Now consider a comparison of the models log2(Income2005) ~ AFQT + Educ and log2(Income2005) ~ AFQT. Why might it be prudent to apply a log-transformation to these data when seeking to statistically evaluate the role of the predictor Educ? Hint: Think about the diagnostic plots you created in Part G.
• Part I: Check same diagnostic plots you created in Part G for the model log2(Income2005) ~ AFQT + Educ. Do the assumptions that underly the \(F\)-test seem more reasonable or less reasonable after applying a log-transformation to the outcome variable? Briefly explain. Hint: Pay attention to the axes scales in diagnostic plots when judging departures from the assumed conditions.
• Part J (extra credit): For the model log2(Income2005) ~ AFQT + Educ, interpret the adjusted effect of a 1-percentile increase in AFQT on income. Note that you must properly apply an inverse transformation to facilitate an appropriate interpretation.

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Question #2

The “email spam” data set contains roughly 4000 emails received by the Gmail Account of the statistician David Diez in the early months of 2012. Additional details on the data can be found here. This question focuses on the variables defined below:

• spam - the outcome variable, a binary indicator of whether the user considered an email to be spam
• dollar - the number of times a \(\$\) symbol or the word “dollar” appeared in the email.
• exclaim_mess - the number of exclaimation points, or ! symbols, that appeared in the email.

emails = read.csv("https://remiller1450.github.io/data/email_spam.csv")

• Part A: Considering outcome variable, spam, briefly explain why logistic regression is a more appropriate model for these data than linear regression.
• Part B: Fit a logistic regression model using dollar to predict spam. Interpret the estimated intercept of this model. Be sure you exponentiate to facilitate a meaningful interpretation.
• Part C: Now interpret the effect of dollar on the likelihood of an email being spam using the model you fit in Part B. Be sure you exponentiate to facilitate a meaningful interpretation.
• Part D: Fit a logistic regression model using both dollar and exclaim_mess to predict the likelihood of an email being spam. Interpret the adjusted effect of dollar in this model. Why might the adjusted effect differ from the marginal effect (which you found in Part C)? Briefly explain.