In estimating the mean of a population, how does the probable accuracy of the estimate improve?

The probable accuracy of estimating the mean of a population improves significantly with an increase in the sample size, denoted as ( n ). This principle stems from the concept of the Central Limit Theorem, which states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution, provided the sample size is sufficiently large.

Moreover, as the sample size grows, the standard error of the mean, which quantifies how much the sample mean is expected to vary from the true population mean, decreases. The standard error is calculated by dividing the population standard deviation ( s ) by the square root of the sample size ( n ). Therefore, increasing ( n ) effectively narrows the confidence intervals around the estimate of the mean, leading to more precise and accurate results.

This is why an increase in ( n ) directly correlates with improved accuracy in estimating the population mean. Larger samples tend to provide more reliable outcomes, diminishing the influence of outliers and variability present in smaller samples, which enhances the robustness of the estimate.