Final answer:
The means of the samples vary with sample size such that larger samples lead to sample means that are more representative of the population mean, with reduced variability and narrower confidence intervals due to the Central Limit Theorem.
Step-by-step explanation:
The means of the samples vary with the sample size because as the sample size increases, the sample mean tends to get closer to the theoretical mean (population mean, µ), and the standard deviation of the sample means (often referred to as the standard error) tends to decrease.
This occurs due to the Central Limit Theorem, which states that as sample size increases, the sampling distribution of the sample mean becomes more normally distributed and centers around the population mean. Additionally, when the sample size increases, the confidence intervals for estimating the population mean become narrower, indicating a more precise estimate.
As noted in the provided information, for smaller samples, there is more variability in the sample means and wider confidence intervals; hence, the estimates of the population mean are less precise. Larger samples provide more reliability and a better statistical estimate as sample results better approximate the actual population average due to reduced sampling variability.
Therefore, as the sample size changes, the shape of the distribution of sample means becomes more symmetrical and narrow, centering around the population mean, and the confidence intervals become smaller, reflecting greater precision in estimation.