Sta-209 (Fall 2026) Homework #4

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

A previous assignment introduced the “infant heart” dataset, which comes from an experiment conducted by esearchers at Harvard Medical School. Infants born with congenital heart defects were randomly assigned to receive one of two different open-heart surgeries, low-flow bypass or circulatory arrest. Two years after the surgery the researchers followed up on each infant and assessed their development, measured by their MDI (mental development index) and PDI (psychomotor development index) scores. The data from this experiment are loaded into R by the code provided below:

ih_data = read.csv("https://remiller1450.github.io/data/InfantHeart.csv")

• Part A: Create a scatterplot of the two primary outcomes, MDI and PDI, and add a LOESS smoother to your scatterplot.
• Part B: Qualitatively interpret the relationship between MDI and PDI using the scatterplot you created in Part A. Be sure to comment on the direction, form, and strength of the relationship (if one is present).
• Part C: Based upon your assessment of the scatterplot created in Part A, would you recommend using Pearson’s or Spearman’s correlation for these data? Briefly explain your rationale.
• Part D: Find and report the correlation coefficient you decided upon in Part C.
• Part E: Suppose a new infant were added to the dataset with an unusually low MDI score but an unusually high PDI score. How might this observation affect the correlation coefficient? Explain your reasoning without attempting to recalculate the correlation.
• Part F: Recall that correlation only measures the strength of a relationship between two variables. Briefly describe one important feature of a scatterplot that the correlation coefficient does not capture particularly well (either in general, or in this specific example).

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Question #2

The data below were collected by The College Scorecard on all primarily undergraduate colleges in the US in the year 2024.

colleges2024 = read.csv("https://remiller1450.github.io/data/Colleges_2024_Complete.csv")

• Part A: Use the select() function (described in Lab 8) to create a subset of the data containing the variables Med_ACT, Enrollment, Avg_Fac_Salary, and Med_Grad_Earnings_4Y.
• Part B: Create a correlation matrix of these variables using Spearman’s correlation. Which pair of variables appears most strongly related?
• Part C: Create a scatterplot matrix of these variables. Do any variables appear to have non-linear or non-monotic relationships?
• Part D: In Part B you should have found that Med_ACT and Enrollment are most weakly related. Suppose we perform a hypothesis test to evaluate whether there is statistical evidence of an association between these variables. Does their weak correlation in the sample data mean that the \(p\)-value of this test will not be statistically significant? Briefly explain your reasoning.
• Part E: Use cor.test() to perform the hypothesis test described in Part D. Report the \(p\)-value as part of a proper 1-sentence conclusion.
• Part F: Spearman’s correlation between Med_ACT and Med_Grad_Earnings_4Y is 0.612. Should this strong, positive correlation be interpreted as suggesting that individuals who score higher on the ACT tend to earn higher incomes four years after they graduate from college? Briefly explain.
• Part G: Spearman’s correlation between Avg_Fac_Salary and Med_Grad_Earnings_4Y is 0.644. Does this strong, positive correlation suggest that if a university wants to increase the earnings of its students after graduate it should increase the salaries of its faculty? Briefly explain.