Finite Population Correction

Definition

The finite population correction is a statistical adjustment made to the formula for calculating the standard error of a sample statistic when the sample size is a significant proportion of the total population size. It accounts for the fact that sampling from a finite population without replacement reduces the variability of the sample compared to sampling from an infinite population.

5 Must Know Facts For Your Next Test

1. The finite population correction is used when the sample size (n) is a significant proportion of the total population size (N), typically when n/N > 0.05 or 5%.
2. The finite population correction reduces the standard error of the sample statistic, resulting in a more accurate estimate of the true population parameter.
3. The formula for the finite population correction is: $\sqrt{\frac{N-n}{N-1}}$, where N is the population size and n is the sample size.
4. The finite population correction is particularly important in the context of the Central Limit Theorem, as it affects the shape and variability of the sampling distribution of the sample mean.
5. In the context of 7.4 Central Limit Theorem (Pocket Change), the finite population correction may be relevant if the total number of coins in the population (e.g., all the coins in the pockets of students on campus) is relatively small compared to the sample size used to study the distribution of pocket change.

Review Questions

• Explain the purpose of the finite population correction and how it differs from sampling from an infinite population.
  • The finite population correction is used to adjust the standard error of a sample statistic when the sample size is a significant proportion of the total population size. This is because sampling from a finite population without replacement reduces the variability of the sample compared to sampling from an infinite population. The finite population correction accounts for this reduced variability, resulting in a more accurate estimate of the true population parameter. In contrast, when sampling from an infinite population, the variability of the sample statistic is not affected by the population size, and the finite population correction is not necessary.
• Describe the relationship between the finite population correction and the Central Limit Theorem.
  • The finite population correction is particularly important in the context of the Central Limit Theorem, which states that the sampling distribution of the sample mean will approach a normal distribution as the sample size increases. The finite population correction affects the shape and variability of the sampling distribution of the sample mean. Specifically, the finite population correction reduces the standard error of the sample mean, resulting in a narrower and more peaked sampling distribution compared to sampling from an infinite population. This is relevant in the 7.4 Central Limit Theorem (Pocket Change) topic, as the finite population correction may need to be considered if the total number of coins in the population (e.g., all the coins in the pockets of students on campus) is relatively small compared to the sample size used to study the distribution of pocket change.
• Analyze the conditions under which the finite population correction should be applied and explain the implications for statistical inference.
  • The finite population correction should be applied when the sample size (n) is a significant proportion of the total population size (N), typically when n/N > 0.05 or 5%. In such cases, the finite population correction reduces the standard error of the sample statistic, resulting in a more accurate estimate of the true population parameter. This has important implications for statistical inference, as it affects the width of confidence intervals and the power of hypothesis tests. When the finite population correction is applied, the confidence intervals will be narrower, and the statistical tests will have higher power to detect differences or effects, compared to the case where the finite population correction is not applied. Failing to account for the finite population correction when it is appropriate can lead to overly conservative or liberal conclusions, underscoring the importance of considering this adjustment in statistical analyses.

Related terms

Standard Error: The standard error is a measure of the variability or spread of a sample statistic, such as the sample mean or proportion, around the true population parameter.

Sampling without Replacement: Sampling without replacement means that once an item is selected from the population, it is not returned to the population before the next selection is made, resulting in a smaller population size for subsequent selections.

Central Limit Theorem: The Central Limit Theorem states that the sampling distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the shape of the original population distribution.