Relative to the population mean, what happens to the possible sample means for samples of the same size as the sample size increases? Explain the relevance of this property in estimating a population means by a sample mean.

The possible sample means cluster increasingly closely around the population mean as the sample size grows. That is, sample means for a higher sample size are less deviated from the population mean than sample means for a smaller sample size. As a result, as the sample size grows, the standard deviation of the sample means reduces, because the standard deviation of a set of observations is actually the root mean square departure of the observations from the mean value.

We calculate the population mean using the sample mean. i.e., we pick a sample at random from all feasible samples of a given size and use the sample mean to estimate the population mean. So, if sample means are close to the population mean for a particular sample size (i.e., the S.D. of sample means is small), we are more likely to achieve a decent estimate of the population mean, i.e., we estimate the population mean with less sampling error. As a result, when the population is estimated using a sample mean with a larger sample size, one should expect a good estimate.