Final answer:
The sample sizes necessary for various population sizes with a certain confidence interval depend on the desired confidence level and margin of error. Larger populations have a diminishing effect on determining sample size, with a 95% confidence level being common. As the confidence level increases, the required sample size increases as well, especially for smaller populations.
Step-by-step explanation:
The sample sizes necessary for population sizes of 1 billion, 10,000, and 100 with a specific confidence interval depend on a variety of factors, including the desired confidence level and margin of error. When determining sample sizes, it is essential to consider the confidence level, which indicates the degree of certainty that the sample reflects the population within a certain error margin. The population size itself has a diminishing effect on sample size as it increases, especially when dealing with populations much larger than the sample.
For large populations, such as 1 billion, the sample size does not need to be proportionally large to achieve a high level of accuracy, thanks to the Central Limit Theorem which implies that for a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the population distribution. A commonly used confidence level is 95%, which is associated with a certain z-score or t-score number that then helps determine the required sample size for the given confidence.
Moreover, the law of large numbers states that as the sample size increases, the sample mean will be closer to the population mean, which aids in achieving a more precise estimate of the population parameter. However, as the desired confidence level increases, so does the required sample size to maintain the desired precision of the estimate. For smaller populations, like 100 individuals, the sample size will represent a larger fraction of the population, and thus has a smaller margin for error to achieve the same confidence as with larger populations.