You use the finite population correction factor when
A) you sample without replacement and the sample size is larger than 5% of the population size.
B) you sample without replacement and the sample size is smaller than 5% of the population size.
C) you sample with replacement and the sample size is larger than 5% of the population size.
D) you sample with replacement and the sample size is smaller than 5% of the population size.
The correct answer and explanation is :
The correct answer is:
A) You sample without replacement and the sample size is larger than 5% of the population size.
Explanation:
The finite population correction (FPC) factor is used to adjust the standard error of a sample statistic when sampling is done without replacement from a finite population. This correction accounts for the reduction in variability due to a relatively large sample size in comparison to the total population.
The FPC factor is applied when the sample size, n, is more than 5% of the total population size, N (i.e., n/N > 0.05). The formula for the correction factor is:
[
FPC = \sqrt{\frac{N – n}{N – 1}}
]
This correction is necessary because when a sample is drawn without replacement, each selection reduces the number of remaining elements, affecting the overall variability. When the sample size is small compared to the population (less than 5%), the effect of this reduction is negligible, and the standard error remains nearly the same as in infinite population scenarios.
However, when the sample size is relatively large (greater than 5% of the population), failing to apply the FPC would overestimate the standard error, leading to unnecessary caution in statistical inference.
Why Other Options Are Incorrect:
- B) Incorrect: The correction is not needed when the sample size is small (less than 5%) because its effect on the variability is minimal.
- C) Incorrect: When sampling with replacement, each draw is independent, and the population size does not impact the variance, so no correction is needed.
- D) Incorrect: Similar to (C), with replacement sampling does not require the FPC factor.
