Sta-209 (Fall 2025) Homework #8

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

A commonly referenced example of Simpson’s Paradox involves the batting averages of professional baseball players. In 1995, Derek Jeter got a hit at 12 of 48 at bats (0.250 batting average) while David Justice got a hit at 104 of 411 at bats (0.253 batting average). In 1996, Derek Jeter got a hit at 183 of 582 at bats (0.314 batting average), while David Justice got a hit at 45 of 140 at bats (0.321 batting average). Thus, David Justice had a higher batting average than Derek Jeter in both of these two seasons.

• Part A: Combining data from both seasons, which player had the higher batting average? Why is this result considered paradoxical when it is compared to the conditional outcomes (that condition on the season)? Briefly explain.
• Part B: Using the example calculation in the “Adjusted Effects” section of Lab 13 as a template, show that Derek Jeter’s overall batting average (combined across both seasons) equals a weighted sum of his batting averages in each season.
• Part C: Following the example calculation in Lab 13, adjust Derek Jeter’s marginal effect (overall batting average combined across both seasons) so that his distribution of at bats each year matches the distribution of David Justice’s at bats each year (ie: use David Justice as a reference). Show your calculation.

\(~\)

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: Create a scatter plot showing the relationship between the explanatory variable height and the response variable IOCT_Time that includes both linear and moving average smoothers. Briefly explain why Pearson’s correlation coefficient is appropriate for measuring the strength of association for these data.
• Part C: 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 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 D and E, state whether or not you believe sex confounds the relationship between height and IOCT_Time.
• Part G: Use the group_by() and summarize() functions in the dplyr package to perform a stratified analysis that reports the correlation between height and IOCT_Time separately for each category of sex. Hint: If you are specifying the data as the first step in the pipeline you should reference the variables height and IOCT_Time without using the $ operator inside of the summarize() function.
• Part H: Briefly interpret the results of the stratified analysis you performed in Part G. That is, does height appear associated with IOCT_Time after you account for the variable sex?
• Part I: Fit the multivariable linear regression model IOCT_Time ~ height + sex and interpret the slope coefficient that describes the effect of changes in height on the expected completion time in the fitted model. Briefly explain the difference between this effect and the one from Part C.

\(~\)

Question #3

For this question you’ll work with the “2023 Boston Marathon” data set found below:

marathon = read.csv("https://data.scorenetwork.org/data/boston_marathon_2023.csv")

This data set records information on each finisher of the 2023 Boston Marathon, with the variable finish_net_sec (finishing time in seconds) being the outcome of interest for this question.

• Part A: Create a frequency table of the variable age_group and briefly describe the distribution of finisher ages.
• Part B: Fit the regression model finish_net_sec ~ age_group. Based upon your fitted model, which age group has the fastest expected finish time?
• Part C: Briefly interpret the estimated coefficient for “age_group50-54” in the model you fit in Part B.
• Part D: Create a data visualization showing the relationship between half_time_sec (the runner’s time at the race’s halfway point) and finish_net_sec. Using this visualization, briefly describe the relationship between these two variables.
• Part E Fit the regression model finish_net_sec ~ half_time_sec. Briefly interpret the slope and intercept of this model. If the intercept is not meaningful, you should indicate this.
• Part F: Now fit the multivariable regression model finish_net_sec ~ half_time_sec + age_group. Explain why the coefficient of “age_group50-54” in this model is so different from the coefficient in the model you fit in Part B.