Final answer:
The sampling distributions and the Central Limit Theorem, which describes the behavior of the sample mean and its tendency to form a normal distribution as sample size increases.
Step-by-step explanation:
The concept of sampling distributions in statistics, specifically dealing with the estimation of population variance using a sample. When samples of size n are drawn from a normal distribution with a true mean (μ) and variance (σ2), as n increases, the distribution of the sample mean (X) approaches a normal distribution, according to the Central Limit Theorem. This theorem states that the mean of the sampling distribution of the mean is the population mean (μ), and its standard deviation (σ) is the population standard deviation (σ) divided by the square root of the sample size (n), denoted as σ/√n.
The empirical rule or 68-95-99.7 rule, relevant to bell-shaped curves, also applies here; it suggests that approximately 95% of sample means will fall within two standard deviations of the population mean. Therefore, we can calculate the range in which we would expect 95% of sample means to lie. In the context of an exam with μ = 81 and σ = 15, the sample mean is likely to fall within two standard deviations (±2*15) of the population mean with 95% confidence.