Public vs. Private Colleges: Comparing Graduation Rates with a Two-Sample t Procedure
Students take a random sample of 40 public and 40 private colleges, compare six-year graduation rates with boxplots, and run a two-sample t test and interval. Then they check the answer against all 1,081 colleges.
Learning objectives
- Compare two distributions with parallel boxplots, using center, spread, shape and outliers in context.
- State hypotheses, check conditions and carry out a two-sample t test for a difference in means.
- Construct and interpret a 95% confidence interval for a difference in population means.
- Explain what a confidence interval captures by comparing it with the known population difference.
Materials
- The lesson CSV (rows with in_class_sample = 1 are the class sample)
- A graphing calculator or statistics software (2-SampTTest and 2-SampTInt)
- The student worksheet below
CSV: https://collegecalcai.com/api/data-csv/teach/public-vs-private-graduation
About the data: Six-year graduation rate of first-time, full-time students: College Scorecard (C150_4), release 2026-06-10, rounded to a whole percent. The class sample was drawn at random (fixed seed): 40 public and 40 private colleges.
Columns: college, state, type, grad_rate_6yr, in_class_sample.
Student worksheet
Name: ______________________ Period: ______ Open the CSV in a spreadsheet, a graphing calculator or statistics software.
- Filter to in_class_sample = 1. Make parallel boxplots of grad_rate_6yr by type and compare the distributions in context.
- Compute the sample mean, standard deviation and size for each type.
- State hypotheses for testing whether the mean six-year graduation rate differs between private and public colleges. Check the conditions.
- Carry out the test. Report the test statistic, degrees of freedom and P-value, and state a conclusion at the 5% level in context.
- Construct and interpret a 95% confidence interval for the difference in mean graduation rates (private minus public).
- Now use every row. Compute the true difference in mean graduation rates. Did your interval capture it?
Teacher key
Every value below is computed by code from the same CSV students download.
- Answers vary with the plots; compare medians, IQRs and outliers in context. The private sample's mean is 62.0% and the public sample's is 56.3%.
- Private: mean 61.98, s = 21.90, n = 40. Public: mean 56.33, s = 17.28, n = 40.
- H0: mu_private = mu_public; Ha: mu_private is not equal to mu_public, where each mu is the true mean six-year graduation rate for that type of college in the dataset. Random: the class sample was chosen at random. 10%: 40 is less than 10% of the 624 private colleges and 40 is less than 10% of the 457 public colleges. Normal/Large Sample: both samples have n of at least 30.
- t = 1.28, df = 74.0 (calculator, Welch), P-value = 0.2042. Because the P-value is at least 0.05, we fail to reject H0. There is not convincing evidence of a difference in mean six-year graduation rates.
- 5.65 plus or minus 1.993 x 4.411 gives (-3.14, 14.44). We are 95% confident that the interval from -3.14 to 14.44 percentage points captures the true difference in mean six-year graduation rate (private minus public).
- All 624 private colleges average 61.25% and all 457 public colleges average 55.09%, a true difference of 6.15 points. The interval captured it. About 95% of intervals made this way capture the true difference; compare across the class.
Extension
- Every student draws their own random sample of 40 and 40. What fraction of the class's intervals capture the true difference?
- Why is a two-sample procedure, not a paired one, correct here?
Free to use and share (CC BY 4.0). Source: CollegeCalcAI, collegecalcai.com/teach/public-vs-private-graduation, from U.S. Department of Education data (IPEDS and the College Scorecard). Unit names follow the College Board AP Statistics Course and Exam Description; not endorsed by the College Board.