Explanation 1
Sampling fraction: n/N, the share of the population sampled.
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Lesson 15 of 18
Evaluate a binomial approximation using p=K/N and a visible sampling fraction.
Advanced / Optional
Evaluate a binomial approximation using p=K/N and a visible sampling fraction.
Explanation 1
Sampling fraction: n/N, the share of the population sampled.
Explanation 2
Approximate p: Population success share K/N.
Explanation 3
Replacement approximation: Treat dependent draws as nearly independent when the fraction is small.
Terminology
n/N, the share of the population sampled.
20 of 500.
Population success share K/N.
40/500.
Treat dependent draws as nearly independent when the fraction is small.
Binomial comparison.
Notation and formulas
Hypergeometric(N,K,n) approximately Binomial(n,K/N)
Define X, its support, parameters, units, and assumptions before substituting values.
Interactive Mission
Compare exact hypergeometric and approximate binomial mass.
P(X = x)
0.0329
F(x) = P(X <= x)
0.9945
E(X)
1.5
SD(X)
1.0629
| x | P(X=x) | Included in F(4) |
|---|---|---|
| 0 | 0.1766 | Yes |
| 1 | 0.3592 | Yes |
| 2 | 0.2963 | Yes |
| 3 | 0.1295 | Yes |
| 4 | 0.0329 | Yes |
| 5 | 0.005 | No |
| 6 | 0.0005 | No |
| 7 | 0 | No |
| 8 | 0 | No |
| 9 | 0 | No |
| 10 | 0 | No |
Interpretation: Custom PMF, discrete uniform, Bernoulli, or binomial: inspect support and trial structure before choosing.
Approximation safeguard: Treat the comparison model as approximate and inspect its error against the exact model.
Worked example
Scenario
A small audit sample is taken from a large finite lot.
A small sampling fraction supports, but does not guarantee, a close approximation.
R connection
Run and edit the original script, inspect intermediate output, and keep exact and approximate results clearly labelled.
Live R Lab
Compare exact sampling without replacement with a labelled binomial approximation.
Ready to run
Dataset provenance
Original synthetic data generated for STATLAB Academy. No textbook data used.
Interpretation safeguard
Label the binomial result approximate and show the finite sampling fraction.
Reflection
Name the random variable, support, parameters, units, dependence assumptions, and one limitation.
Assistant
Coach me through Binomial Approximation to the Hypergeometric by asking about X, support, parameters, assumptions, probability notation, exact versus approximate status, and interpretation. Do not provide a quiz answer before submission or write my final capstone recommendation.
Exit check
Use conceptual hints before submitting. Full formulas, calculations, and interpretation appear afterward.
Checkpoint
Question 1 of 2. Answered 0/2. Passing score: 70%.
Resource
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