Final answer:
As the sample size increases, the Central Limit Theorem ensures that the sampling distribution of the mean becomes normally distributed, with its standard deviation, the standard error, decreasing as the size increases. The Law of Large Numbers also indicates that the mean of the sampling distribution gets closer to the population mean.
Step-by-step explanation:
When the sample size increases, the sampling distribution of the mean, often denoted as x-bar, tends to become more normally distributed. This phenomenon is described by the Central Limit Theorem, which purports that if you draw random samples of a sufficient size from a population, the distribution of the sample means will approach a normal distribution, regardless of the population's original distribution.
This convergence to normality occurs because the variability of the sampling distribution decreases as the sample size increases. The standard deviation of the sampling distribution of the mean, known as the standard error, is the population standard deviation divided by the square root of the sample size (σ/n¹⁰¹). Thus, with larger samples, this standard deviation becomes smaller, making the distribution of sample means tighter around the population mean μ. This can be described with the equation for the standard error of the mean, where σ is the population standard deviation and n is the sample size (standard deviation = σ/n¹⁰¹).
Furthermore, the Law of Large Numbers supports this by stating that the sample mean will get closer to the population mean as the sample size grows. Therefore, not only does the shape of the sampling distribution become more normal, but the sample mean also becomes a more accurate estimate of the population mean. In essence, the Central Limit Theorem and the Law of Large Numbers together ensure that for large sample sizes, the sampling distribution of the mean will mirror a normal distribution with mean equal to the population mean and a smaller, reducing standard error.