Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What happens to the sampling distribution of the mean as sample sizes increase?

1. It becomes more varied

2. It approximates a normal distribution

3. It diverges from the actual population mean

4. It remains unchanged

The correct answer is: It approximates a normal distribution

As sample sizes increase, the sampling distribution of the mean approaches a normal distribution due to the Central Limit Theorem. This theorem states that regardless of the shape of the population distribution, as the sample size grows, the distribution of the sample means will tend to become normal. This convergence occurs because larger samples tend to better capture the characteristics of the population, leading to a mean that is more consistent and stable. As the sample size increases, the variability of the sample means around the population mean decreases, resulting in a tighter and more normal-shaped distribution. This phenomenon is crucial in inferential statistics because it allows for the application of normal probability theory to make inferences about population parameters, even when the original population distribution is not normal. The implications of this behavior are significant in fields that rely on statistical analysis, leading to more reliable conclusions as sample sizes become larger.