In simple random sampling of n units from a population of N units the quantity [1 - (n / N)] is called the
finite population correction factor
Understanding the Quantity 1 - (n / N) in Simple Random Sampling
The question asks to identify the term for the quantity \(\left[1 - \left(\frac{n}{N}\right)\right]\) in the context of simple random sampling of \(n\) units from a population of \(N\) units. This quantity plays a crucial role in survey sampling, particularly when dealing with finite populations. Let's examine the given options.
Exploring the Options
- Sampling Fraction: The sampling fraction is defined as the ratio of the sample size (\(n\)) to the population size (\(N\)). It is represented as \(\frac{n}{N}\). This term indicates the proportion of the population that is included in the sample.
- Expansion Factor: The expansion factor, also known as the raising factor or weighting factor, is the reciprocal of the sampling fraction. It is defined as \(\frac{N}{n}\). This factor is often used to extrapolate sample results to the entire population; for example, to estimate a population total from a sample total.
- Finite Population Correction Factor: The finite population correction factor, often abbreviated as FPC, is a term used in the calculation of the variance of estimates (like the sample mean or total) when sampling without replacement from a finite population. When sampling without replacement from a finite population, the variance is smaller than it would be if sampling with replacement or from an infinite population. The FPC adjusts the variance formula to account for this. The FPC is defined as \(\left[1 - \left(\frac{n}{N}\right)\right]\) or equivalently \(\left(\frac{N-n}{N}\right)\).
- Degrees of Freedom: Degrees of freedom is a concept used in statistical inference, particularly in relation to variance estimates and probability distributions (like the t-distribution or chi-squared distribution). In the context of a simple random sample of size \(n\), the degrees of freedom for estimating the population variance is typically \(n-1\). It represents the number of values in a calculation that are free to vary.
Identifying the Correct Term
Comparing the quantity given in the question, \(\left[1 - \left(\frac{n}{N}\right)\right]\), with the definitions above, it exactly matches the definition of the finite population correction factor. This factor is essential for accurately estimating the precision of survey results when the sample size is a significant proportion of the population size.
For instance, the variance of the sample mean (\(\bar{y}\)) in simple random sampling without replacement is given by the formula:
\(Var(\bar{y}) = \frac{S^2}{n} \times \left(\frac{N-n}{N}\right) = \frac{S^2}{n} \times \left[1 - \left(\frac{n}{N}\right)\right]\)
where \(S^2\) is the population variance. Here, the term \(\left[1 - \left(\frac{n}{N}\right)\right]\) is the finite population correction factor. If the population is infinite (\(N \to \infty\)) or sampling is done with replacement, the sampling fraction \(\frac{n}{N}\) approaches 0, and the FPC approaches 1. In such cases, the variance formula simplifies to \(\frac{S^2}{n}\).
Summary of Terms
| Term | Formula | Description |
|---|---|---|
| Sampling Fraction | \(\frac{n}{N}\) | Proportion of population sampled. |
| Expansion Factor | \(\frac{N}{n}\) | Reciprocal of sampling fraction, used for extrapolation. |
| Finite Population Correction Factor (FPC) | \(\left[1 - \left(\frac{n}{N}\right)\right]\) or \(\left(\frac{N-n}{N}\right)\) | Factor to adjust variance when sampling without replacement from a finite population. |
| Degrees of Freedom | e.g., \(n-1\) | Number of values free to vary in a calculation. |
Based on the definitions, the quantity \(\left[1 - \left(\frac{n}{N}\right)\right]\) is correctly identified as the finite population correction factor.
Revision Table: Simple Random Sampling Concepts
| Concept | Formula | Importance in Simple Random Sampling |
|---|---|---|
| Sample Size (\(n\)) | \(n\) | Number of units selected from the population. |
| Population Size (\(N\)) | \(N\) | Total number of units in the population. |
| Sampling Fraction | \(\frac{n}{N}\) | Indicates intensity of sampling. |
| Expansion Factor | \(\frac{N}{n}\) | Used for estimating population totals/aggregates from sample data. |
| Finite Population Correction Factor (FPC) | \(\left[1 - \left(\frac{n}{N}\right)\right]\) | Adjusts variance calculations for sampling without replacement from finite populations. Becomes significant when \(n/N\) is large. |
Additional Information: Why the FPC is Needed
When you sample without replacement from a finite population, as you select units for your sample, the composition of the remaining population changes. This means subsequent selections are not independent in the same way they would be with replacement sampling or sampling from an infinite population. Specifically, if you draw a sample of size \(n=N\) (a census), your sample mean will be exactly equal to the population mean, and the variance of the sample mean will be zero. The finite population correction factor reflects this reduction in variance. When \(n\) is much smaller than \(N\) (e.g., \(n/N < 0.05\)), the FPC is close to 1, and the distinction between sampling with and without replacement becomes negligible in terms of variance. However, when \(n\) is a substantial portion of \(N\), the FPC significantly reduces the estimated variance, providing a more accurate measure of precision for estimates obtained via simple random sampling without replacement.
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