Finite Population Correction Factor

Definition

The finite population correction factor is a statistical adjustment applied when sampling from a population that is small relative to the overall size of the population. It accounts for the fact that sampling without replacement from a finite population reduces the variability of the sample compared to sampling with replacement from an infinite population.

5 Must Know Facts For Your Next Test

1. The finite population correction factor is used to adjust the standard error of the sample mean and sample proportion when the population size is small compared to the sample size.
2. The formula for the finite population correction factor is $\sqrt{\frac{N-n}{N-1}}$, where $N$ is the population size and $n$ is the sample size.
3. Applying the finite population correction factor reduces the standard error of the sample mean and sample proportion, leading to a narrower confidence interval and increased precision of the estimates.
4. The finite population correction factor is most relevant when the population size is less than 20 times the sample size, as the impact on the standard error becomes negligible for larger populations.
5. Ignoring the finite population correction factor when it is applicable can lead to overly conservative confidence intervals and hypothesis tests, potentially resulting in incorrect conclusions.

Review Questions

• Explain the purpose of the finite population correction factor and how it relates to sampling from a finite population.
  • The finite population correction factor is used to adjust the standard error of the sample mean and sample proportion when the population size is small compared to the sample size. This is necessary because sampling without replacement from a finite population reduces the variability of the sample compared to sampling with replacement from an infinite population. By applying the finite population correction factor, the standard error is reduced, leading to narrower confidence intervals and increased precision of the estimates. This adjustment is most relevant when the population size is less than 20 times the sample size, as the impact on the standard error becomes negligible for larger populations.
• Describe the formula for the finite population correction factor and explain how it is used to adjust the standard error.
  • The formula for the finite population correction factor is $\sqrt{\frac{N-n}{N-1}}$, where $N$ is the population size and $n$ is the sample size. This formula is used to multiply the standard error of the sample mean or sample proportion calculated under the assumption of an infinite population. By applying this correction factor, the standard error is reduced, reflecting the decreased variability in the sample due to sampling without replacement from a finite population. The adjusted standard error leads to narrower confidence intervals and more precise estimates of the population parameter.
• Discuss the implications of ignoring the finite population correction factor when it is applicable, and explain the potential consequences on the statistical inferences drawn from the data.
  • Ignoring the finite population correction factor when it is applicable can lead to overly conservative confidence intervals and hypothesis tests, potentially resulting in incorrect conclusions. By not applying the finite population correction factor, the standard error of the sample mean or sample proportion will be larger than necessary, leading to wider confidence intervals and a higher likelihood of failing to reject the null hypothesis when it is false (Type II error). This can result in missed opportunities to detect significant differences or relationships in the data, as the statistical tests will have less power due to the inflated standard error. Failing to account for the finite population correction factor when it is relevant can thus compromise the validity and reliability of the statistical inferences drawn from the data.

Related terms

Sampling with Replacement:

A sampling method where each item selected from the population is returned before the next item is selected, allowing the same item to be selected multiple times.

Sampling without Replacement: A sampling method where each item selected from the population is not returned before the next item is selected, ensuring that no item can be selected more than once.

Finite Population: A population that has a known, fixed size, in contrast to an infinite population where the size is unknown or unbounded.