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Sampling Distributions with Real Colleges: Sample Means and Proportions

The 1,013 colleges in the file are the population. Students draw random samples of 25 colleges, compare sample means of admit rate to the population mean, and build the sampling distributions of a mean and a proportion.

AP StatisticsUnit 5: Sampling Distributions; Unit 3: Collecting Data
TimeOne to two 50-minute class periods
Common CoreHSS.IC.A.1, HSS.IC.B.4, HSS.ID.A.2
DataCSV, 1,013 rows

To print, use your browser's print command. The site header and footer are left off, and the teacher key starts on a new page, so you can print the worksheet pages alone.

Learning objectives

  • Select a simple random sample from a numbered population using a random number generator.
  • Distinguish a parameter from a statistic and a population distribution from a sampling distribution.
  • Find the mean and standard deviation of the sampling distribution of a sample mean and of a sample proportion, and check the 10% condition.
  • Compare a simulated sampling distribution with the theoretical one.

Materials

  • The lesson CSV (one numbered row per college)
  • A random number generator or calculator randInt
  • A shared class dot plot (board, sticky notes or a shared spreadsheet)

CSV: https://collegecalcai.com/api/data-csv/teach/sampling-distributions

About the data: Admit rate: IPEDS fall 2023, admitted divided by applied, as a percent with one decimal, for every college in the CollegeCalcAI dataset with IPEDS admissions counts. Simulation: 5,000 samples of 25 drawn without replacement with a fixed seed.

Columns: id, college, state, type, admit_rate_pct.

Student worksheet

Name: ______________________ Period: ______ Open the CSV in a spreadsheet, a graphing calculator or statistics software.

  1. Treat the 1,013 colleges as the population. Use a random number generator to choose 25 different ids. Record each college's admit_rate_pct and compute your sample mean.
  2. Is your sample mean a parameter or a statistic? What is the parameter it estimates?
  3. Add your sample mean to the class dot plot. Describe the shape, center and spread of the class's sample means.
  4. Compute the population mean and population standard deviation of admit_rate_pct.
  5. Find the mean and standard deviation of the sampling distribution of the sample mean for samples of 25. Check the 10% condition.
  6. Let p be the proportion of colleges that admit fewer than 50% of applicants. Find p, then the mean and standard deviation of the sampling distribution of the sample proportion for n = 25.
  7. Is the Large Counts condition met for the sample proportion with n = 25? What does that tell you about its shape?

Teacher key

Every value below is computed by code from the same CSV students download.

  1. Answers vary. Worked example with the fixed class sample (ids 10, 123, 156, 196, 201, 261, 358, 410, 420, 445, 466, 525, 543, 627, 637, 656, 665, 675, 703, 827, 895, 909, 969, 1001, 1005): sample mean 77.21%.
  2. A statistic: it describes a sample. It estimates the parameter mu, the mean admit rate of all 1,013 colleges.
  3. In a simulation of 5,000 samples of 25, the sample means centered at 70.32% with standard deviation 4.60 percentage points, roughly symmetric and mound-shaped. 71.8% of the simulated sample means were within 5 points of the population mean. A class's dot plot will be similar but rougher.
  4. mu = 70.27%, sigma = 23.11 percentage points (population standard deviation, dividing by N = 1,013).
  5. Mean of the sample mean = mu = 70.27%. Standard deviation = sigma / sqrt(n) = 23.11 / sqrt(25) = 4.62. The 10% condition holds: 25 is less than 10% of 1,013 (101.3), so sampling without replacement barely changes this (the exact figure, with the finite population correction, is 4.57).
  6. p = 0.1728. Mean of the sample proportion = 0.1728; standard deviation = sqrt(p(1 minus p)/n) = 0.0756. The simulation gave a mean of 0.1723 and a standard deviation of 0.0756.
  7. n p = 4.32 and n(1 minus p) = 20.68. At least one is below 10, so the sampling distribution of the sample proportion is not approximately normal at this sample size; it is skewed. A larger sample would be needed.

Extension

  • Repeat with samples of 100. Predict the new standard deviation (sigma / 10 = 2.31) before drawing.
  • Make a histogram of the population itself. Explain why the sampling distribution of the mean is closer to normal than the population.

All free lessons

Free to use and share (CC BY 4.0). Source: CollegeCalcAI, collegecalcai.com/teach/sampling-distributions, 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.