توضیح 1
Sampling fraction: n/N, the share of the population sampled.
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درس 15 از 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.
توضیح 1
Sampling fraction: n/N, the share of the population sampled.
توضیح 2
Approximate p: Population success share K/N.
توضیح 3
Replacement approximation: Treat dependent draws as nearly independent when the fraction is small.
اصطلاحات
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.
نمادگذاری و فرمولها
Hypergeometric(N,K,n) approximately Binomial(n,K/N)
Define X, its support, parameters, units, and assumptions before substituting values.
فعالیت تعاملی
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.
مثال حلشده
سناریو
A small audit sample is taken from a large finite lot.
A small sampling fraction supports, but does not guarantee, a close approximation.
ارتباط با R
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.
بازاندیشی
Name the random variable, support, parameters, units, dependence assumptions, and one limitation.
دستیار
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.
ارزیابی پایانی
منبع
Explore additional discrete-distribution explanations from properly licensed or publisher resources.
دریافتمنبع
Download an original worksheet, formula sheet, or capstone tool for this lesson.
دریافت