A finite population correction factor is needed in computing the standard deviation of the sampling distribution of sample means irrespective of the size of the sample. whenever the sample size is more than \( 5 \% \) of the population size. whenever the population is infinite. whenever the sample size is less than \( 5 \% \) of the population size.

The finite population correction factor (FPC) becomes essential when the sample size is significant relative to the population size. Specifically, if the sample size exceeds \( 5 \% \) of the total population, the variance of the sampling distribution should be adjusted to account for the reduced variability in the sample. This correction helps ensure that the standard deviation calculations more accurately reflect the true variability of the population. In practical scenarios, using the FPC can significantly impact the results of survey sampling or quality control processes. For instance, if only a small fraction (less than \( 5 \% \)) of a city’s population is surveyed for their opinions, the sample's standard deviation can be computed without worry. However, if the number of respondents approaches a sizable chunk of the population, applying the FPC becomes crucial for effective analysis and decision-making.