- The finite population correction (FPC) factor is used when sampling without replacement from a finite population.
- It is considered negligible when the sample size \(n\) is small relative to the population size \(N\).
- The standard statistical rule of thumb states that the FPC is not necessary when the sampling fraction \(\frac{n}{N} \leq 0.05\).
- The correct answer is \(\boxed{a}\).
Explanation
- Understand the finite population correction (FPC) factor
The standard error of the mean (SEM) measures how much the sample mean of the population is expected to vary from the actual population mean. When we sample without replacement from a finite population of size \(N\), the standard deviation of the sampling distribution is adjusted by the finite population correction (FPC) factor:
\[\text{FPC} = \sqrt{\frac{N-n}{N-1}}\]
where \(n\) is the sample size. The purpose of this factor is to reduce the standard error as the sample size approaches the population size, reflecting the increased precision gained when sampling a larger portion of a finite population.
- Explain the rule of thumb for using the FPC factor
In statistical practice, applying the FPC factor is often unnecessary if the sample size \(n\) is small relative to the population size \(N\). A widely accepted rule of thumb in statistics is that if the sampling fraction, defined as \(\frac{n}{N}\), is less than or equal to \(0.05\) (or \(5\%\)), the adjustment is considered negligible, and researchers typically omit it. This simplifies calculations without significantly impacting the accuracy of the result.
- Evaluate the options and identify the correct condition
We evaluate the provided options against the rule \(\frac{n}{N} \leq 0.05\):
- Option (a): \(\frac{n}{N} \leq 0.05\) matches the standard rule of thumb.
- Option (b): \(\frac{N}{n} \leq 0.05\) is incorrect as this would imply a very large population relative to a very small sample, which is not the correct criterion.
- Option (c): \(n \geq 30\) refers to the Central Limit Theorem for the distribution of the sample mean but is not the criterion for applying the FPC.
- Option (d): \(\frac{n}{N} > 0.05\) is the condition under which the FPC should be used, as the sample represents a significant portion of the population.
Therefore, the finite population correction factor is not used when \(\frac{n}{N} \leq 0.05\).
Examples
If a school administrator wants to estimate the average reading score of 2,000 students by surveying 50 students, the sampling fraction is \(\frac{50}{2000} = 0.025\). Since \(0.025 \leq 0.05\), the administrator does not need to apply the finite population correction factor when calculating the standard error of the mean, making the statistical analysis more straightforward.