Texas A&M University (TAMU) MATH140 Mathematics for Business and Social Sciences Final Practice Exam

For the sampling distribution of sample means to converge to a normal distribution, the sample size must increase. This concept is supported by the Central Limit Theorem, which states that as the sample size increases, the distribution of the sample means will approach a normal distribution, regardless of the shape of the original population distribution, provided that the samples are sufficiently large and taken from a population with a finite mean and variance.

An increase in sample size reduces the standard error of the mean, which means that sample means will be less spread out and more concentrated around the true population mean. With larger sample sizes, the variability decreases, making the sample mean behave more like a normal distribution even if the original population distribution is skewed or non-normal.

In contrast, the population size does not directly affect the shape of the sampling distribution of the means. Variance is a measure of dispersion within a dataset and may influence the spread of data points but does not directly shape the distribution into a normal form. Similarly, the confidence level relates to statistical inference regarding the precision of estimates, rather than the convergence of the sampling distribution to a normal form. Consequently, increasing the sample size is the critical factor for achieving a normal approximation in the sampling distribution of sample means.