Final answer:
The margin of sample error is independent of the population size and is affected by the sample size, with larger samples leading to a smaller margin of error. As the sample size grows, the sample mean approaches the population mean, and confidence intervals become more precise.
Step-by-step explanation:
In most cases, the margin of sample error of a random sample is independent of the size of the population. This is a fundamental concept in statistics which emphasizes that the accuracy of a sample's reflection of the overall population is not affected by the size of the population itself, but rather the size of the sample and the sampling method employed.
Larger samples tend to have a smaller margin of error, as they more accurately represent the population. The precision of estimating the population mean increases with larger samples due to the law of large numbers, which indicates that as sample size increases, the sample mean will more closely approximate the population mean. Additionally, confidence intervals also become more reliable; for example, if we are considering a 90% confidence interval, about 90% of such calculated intervals from repeated samples would contain the true population mean.
The margin of error also depends on the chosen confidence level and the variability within the data. If a sample was taken with a smaller size, it would be less likely to form a normal distribution, making our approximation less accurate compared to larger samples.