[Sampling] Population Size (Non-replacement Sampling from a Finite Population)
Author: RumexJaponicusExecutions: 524
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In a non-replacement sampling from a finite population, determine the “population size” that satisfies the specified conditions.
If a non-replacement sampling of “sample size” is repeated infinitely many times from a population with a “population size” and a “population proportion,” the “sample proportion” will, with the probability specified by the “confidence level,” approximately satisfy the following condition:
| Sample proportion - Population proportion | ≤ Margin of error
| Sample proportion - Population proportion | ≤ Margin of error
| Population size | |
| Error Code | |
| Estimated proportion of the population | |
| Estimated error limits | |
| Estimated confidence level |
[Error Codes]
Error Code Description
0 Terminated normally.
101 (Sample size < Population size) is not true.
102 (1 / Population size ≤ Population proportion ≤ 1 - 1 / Population size) is not true.
103 ( 3 ≤ Population Size ∧ Population Size ∈ N ) is not true.
201 ( 0 < population proportion - margin of error ∧ population proportion + margin of error < 1 ) is not true.
301 ( 2 ≤ sample size ∧ sample size ∈ N ) is not true.
302 ( 0 < confidence level < 1 ) is not true.
303 ( 0 < margin of error < 0.5 ) is not true.
304 ( 0 < population proportion < 1 ) is not true.
* N is the set of all natural numbers
[Evaluation of Calculation Results]
When the “Error Code” is 0, the calculation results are evaluated based on the hypergeometric distribution. In this evaluation, optimal values are estimated in the order of “population proportion,” “margin of error,” and “confidence level.”
Estimated population proportion P
Estimated margin of error e
Estimated confidence level A
Here,
P = M / Population Size
M = [Population Size * Population Proportion + 0.5]
e = max(|e_min|, |e_max|)
e_min = m_min / Sample Size - P
e_max = m_max / sample size - P
m_min = [sample size * (P - margin of error) + 0.5]
m_max = [sample size * (P + margin of error) + 0.5]
A = 1 - (B_lower + B_upper)
Sample size
B_lower = 1 - Σ H(x; population size, sample size, P)
x = m_min
m_max
B_upper = 1 - Σ H(x; population size, sample size, P)
x = 0
where H(x; N, n, p) is the probability mass function of the hypergeometric distribution.
Error Code Description
0 Terminated normally.
101 (Sample size < Population size) is not true.
102 (1 / Population size ≤ Population proportion ≤ 1 - 1 / Population size) is not true.
103 ( 3 ≤ Population Size ∧ Population Size ∈ N ) is not true.
201 ( 0 < population proportion - margin of error ∧ population proportion + margin of error < 1 ) is not true.
301 ( 2 ≤ sample size ∧ sample size ∈ N ) is not true.
302 ( 0 < confidence level < 1 ) is not true.
303 ( 0 < margin of error < 0.5 ) is not true.
304 ( 0 < population proportion < 1 ) is not true.
* N is the set of all natural numbers
[Evaluation of Calculation Results]
When the “Error Code” is 0, the calculation results are evaluated based on the hypergeometric distribution. In this evaluation, optimal values are estimated in the order of “population proportion,” “margin of error,” and “confidence level.”
Estimated population proportion P
Estimated margin of error e
Estimated confidence level A
Here,
P = M / Population Size
M = [Population Size * Population Proportion + 0.5]
e = max(|e_min|, |e_max|)
e_min = m_min / Sample Size - P
e_max = m_max / sample size - P
m_min = [sample size * (P - margin of error) + 0.5]
m_max = [sample size * (P + margin of error) + 0.5]
A = 1 - (B_lower + B_upper)
Sample size
B_lower = 1 - Σ H(x; population size, sample size, P)
x = m_min
m_max
B_upper = 1 - Σ H(x; population size, sample size, P)
x = 0
where H(x; N, n, p) is the probability mass function of the hypergeometric distribution.
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