Sta-209 (Fall 2026) Homework #7

Directions:

• Submit your final answers and all supporting work on Canvas (under the “Assignments” tab)
• You must submit HTML output generated using R Markdown for this assignment.
• Homework is intended to be individual work. While you may discuss the assignment with your peers, you should submit answers that are uniquely your own.
• Any assistance you receive from resources/materials not on our course website (such as other websites, course mentors, peer students, AI, etc.) should be clearly acknowledged.

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.

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

The data provided above are a random sample of \(n=2584\) Americans who took the AFQT in 1979, then were interviewed in 2006 about their income and educational attainment (number of years of formal education).

• Part A: Create a scatter plot depicting the relationship between the predictor AFQT (x-axis) and the outcome Income2005 (y-axis). Do these variables appear related? If so, describe the direction, strength, and form of the relationship.
• Part B: Fit a simple linear regression using AFQT to predict Income2005. Print a summary of the model and interpret the fitted slope coefficient.
• Part C: Suppose higher AFQT scores are associated with higher incomes and higher amounts of education. If higher educational attainment is also associated with a higher income, do you expect the marginal effect of AFQT on income to be larger or smaller than the adjusted effect (adjusting for differences in education)? Briefly explain your reasoning.
• Part D: Now fit a multivariable regression model using both AFQT and Educ to predict Income2005. Print a summary of this model and interpret the fitted slope coefficient of AFQT.
• Part E: Suppose you are a researcher trying to separate the effects of education and intelligence on income. Is the marginal effect (from Part B) or the adjusted effect (from Part D) more useful in this situation? Briefly explain.

\(~\)

Question #2

The Indoor Obstacle Course Test (IOCT) is an obstacle course that all Cadets at West Point Academy are tested on and must successfully complete prior to graduation. The data set below records IOCT completion times for 384 graduates of the academy:

ioct = read.csv("https://data.scorenetwork.org/data/ioct_west_point.csv")

• Part A: Find Pearson’s correlation coefficient measuring the strength of association between the variables height and IOCT_Time. Based upon this correlation, do taller Cadets tend to finish the course faster or slower than shorter Cadets? Briefly explain.
• Part B: Fit the simple linear regression model IOCT_Time ~ height and interpret the slope coefficient that describes the effect of changes in height on the expected completion time in the fitted model.
• Part C: Using a printed summary of the model from Part B, perform a statistical test to evaluate whether these data provide evidence of a linear relationship between height and IOCT completion time. Provide a one-sentence conclusion including your \(p\)-value, context, and the direction of the effect (if one was found).
• Part D: Create an appropriate data visualization to determine whether the variables sex and height are associated in these data. Using your visualization, briefly explain whether or not you believe these variables are associated.
• Part E: Create an appropriate data visualization to determine whether the variables sex and IOCT_Time are associated in these data. Using your visualization, briefly explain whether or not you believe these variables are associated.
• Part F: Based upon your answers to Parts B-E, do you expect the adjusted effect of height (after adjusting for differences in sex) to be larger, smaller, or approximately the same as the marginal effect estimated in Part B? Briefly explain your reasoning.
• Part G: Fit the multivariable linear regression model IOCT_Time ~ height + sex and interpret the slope coefficient that describes the effect of height on the expected completion time in the fitted model.
• Part H: Using a printed summary of the model from Part G, perform a statistical test to evaluate whether these data provide evidence of a linear relationship between height and IOCT completion time after adjusting for differences in sex. Provide a 1-sentence conclusion including your \(p\)-value, context, and the direction of the effect (if one was found).
• Part I: Suppose you are selecting four West Point Academy Cadets to compete in a relay-style obstacle course competition. You can visually determine each candidate’s approximate height and sex, but do not have access to any other information. Based upon the findings so far, should your selections be based primarily on sex, primarily on height, or on both variables to a similar extent? Briefly explain your reasoning using evidence from your fitted models.
• Part J: Now fit a final model using the three predictors height, sex, and weight to predict IOCT_Time. Interpret the coefficient of sexM in this model.
• Part K (optional, extra credit for correct answers): Notice the coefficient of height changes substantially depending upon whether sex and weight are included in the model, but the coefficient of sex tends to be similar (between -80 and -90) regardless of which other predictors are in the model. Explain why the estimated effect of height depends so strongly on the other variables included in the model, while the estimated effect of sex changes relatively little.