Final answer:
The Central Limit Theorem (CLT) implies that as the sample size increases, the distribution of sample means becomes normal with the SD of the sum increasing by a factor of √n and SEM decreasing by a factor of 1/√n, reflecting the increased precision in estimating the population mean.
Step-by-step explanation:
The Central Limit Theorem (CLT) has significant implications for the behavior of sample means. It implies that as the size of the sample (n) gets larger, the distribution of sample means approaches a normal distribution, regardless of the initial distribution of the population. A key aspect of this theorem is its influence on the standard deviation (SD) and the standard error of the mean (SEM).
Standard Deviation (SD) is a measure of variability or dispersion of a set of values. According to the CLT, the standard deviation of the sampling distribution of the sum (σx sum) is equal to the population standard deviation (σx) multiplied by the square root of the sample size (n). This relationship is denoted as σx sum = (√n)(σx).
Standard Error of the Mean (SEM), on the other hand, is the standard deviation of the sampling distribution of the sample mean. CLT states that the SEM is equal to the population standard deviation (σx) divided by the square root of the sample size (n), which can be expressed as SEM = σx / √n. As the sample size increases, the SEM decreases, resulting in a more precise estimate of the population mean.
In summary, the Central Limit Theorem ensures that the SD of sample sums and the SEM both depend on the sample size and population standard deviation. Their calculation involves the square root of the sample size (n), highlighting how larger samples lead to more normal and narrower distributions of sample means or sums.