Sample Statistics vs Population Parameters

A sample statistic is a numerical summary derived from a sample (a subset of the population). A population parameter is a numerical summary describing the entire population.

The question notes that the population size is always larger than the sample size. This implies we are using inferential statistics, where we estimate population characteristics using sample data.

Analyzing Sample Statistic Variability

The relationship between a sample statistic and its corresponding population parameter is subject to sampling variability. This means the statistic can fluctuate from sample to sample.

• Option 1: Can never be larger than the population parameter. This is incorrect. For example, a sample mean could be higher than the population mean due to random chance in sample selection.
• Option 2: Can never be smaller than the population parameter. This is also incorrect. A sample mean could be lower than the population mean for the same reason.
• Option 3: Can never be equal to the population parameter. This is incorrect. While often unlikely, it is possible for a sample statistic to exactly equal the population parameter, especially if the sample perfectly mirrors the population or by sheer coincidence.

Conclusion on Correct Answer

Since the first three options make absolute statements ("can never be") that are demonstrably false, none of them accurately describe the potential relationship between a sample statistic and a population parameter.

Therefore, the correct choice is that none of the provided statements (1, 2, or 3) is correct.