When sampling from small populations, the finite population correction factor becomes crucial. It adjusts standard errors and variances when sample size exceeds 5% of the population, improving estimate accuracy for means and proportions.
This factor accounts for reduced variability in larger samples relative to population size. It's essential in scenarios like election polls, quality control, and market research, where sampling without replacement impacts independence and estimation precision.
Finite Population Correction Factor
Finite population correction factor calculation
- Adjusts standard error of sampling distribution when sample size exceeds 5% of population size
- Formula: $\sqrt{\frac{N-n}{N-1}}$ where $N$ = population size and $n$ = sample size
- Apply to standard error of sampling distribution of means by multiplying $\sigma_{\bar{x}} \cdot \sqrt{\frac{N-n}{N-1}}$ where $\sigma_{\bar{x}}$ = standard error of sampling distribution of means
- Apply to standard error of sampling distribution of proportions by multiplying $\sigma_{\hat{p}} \cdot \sqrt{\frac{N-n}{N-1}}$ where $\sigma_{\hat{p}}$ = standard error of sampling distribution of proportions
- Reduces standard error as it accounts for the reduction in variability when a larger proportion of the population is sampled (election polls, small town surveys)
- Used when sampling without replacement from a finite population
Adjustment of sampling variances
- Adjusts variance of sampling distribution when sample size exceeds 5% of population size
- Apply to variance of sampling distribution of means by multiplying $\sigma^2_{\bar{x}} \cdot \frac{N-n}{N-1}$ where $\sigma^2_{\bar{x}}$ = variance of sampling distribution of means
- Apply to variance of sampling distribution of proportions by multiplying $\sigma^2_{\hat{p}} \cdot \frac{N-n}{N-1}$ where $\sigma^2_{\hat{p}}$ = variance of sampling distribution of proportions
- Reduces variance as it accounts for the reduction in variability when a larger proportion of the population is sampled (quality control testing, market research on niche segments)
Independence in population sampling
- Assumes each observation is independent of all other observations in the sample
- Violated when sample size exceeds 5% of population size as probability of selecting an individual changes after each selection due to decreasing population size (drawing cards without replacement, selecting marbles from a jar)
- Population proportion near 0 or 1 also affects independence as probability of selecting an individual with the characteristic of interest changes significantly after each selection, especially with larger sample sizes relative to population (rare disease testing, defect rates in manufacturing)
- Lack of independence can lead to biased estimates and incorrect conclusions about the population (overestimating support for an unpopular policy, underestimating failure rates of a product)
Statistical Inference and Estimation
- Central limit theorem allows for inference about population parameters from sample statistics
- Margin of error quantifies the precision of estimates derived from samples
- Population parameters are estimated using corresponding sample statistics
- Finite population correction factor improves the accuracy of these estimates for small populations