Question

State the Central Limit Theorem. Of what value is the Central Limit Theorem in large-sample statistical estimation?

Short answer

Answer: The Central Limit Theorem states that the distribution of the sum or average of a large number of independent, identically distributed random variables will approach a normal distribution, regardless of the original distribution of the individual variables. Its value in large-sample statistical estimation lies in the fact that it allows for normal approximation, forms the basis of inferential statistics techniques, improves estimator accuracy, and has wide-ranging applicability across disciplines.

Step-by-step solution

State the Central Limit Theorem

The Central Limit Theorem (CLT) states that the distribution of the sum (or average) of a large number of independent, identically distributed random variables will approach a normal distribution, regardless of the original distribution of the individual variables. In mathematical terms, if we have a large number (n) of independent random variables, X_1, X_2, ..., X_n, each with the same mean (μ) and variance (σ^2), then the distribution of the sample mean (X̄) will approach a normal distribution with mean μ and variance σ^2/n as n goes to infinity.

Value of the Central Limit Theorem in large-sample statistical estimation

The Central Limit Theorem is of great value in large-sample statistical estimation due to the following reasons: 1. Normal approximation: Since the CLT states that the distribution of the sample mean approaches a normal distribution for a large sample size, it allows us to use the well-studied properties of the normal distribution for sample means, even when the underlying population distribution is not normal. This simplifies the analysis of many statistical problems significantly. 2. Confidence intervals and hypothesis testing: The CLT forms the basis of various inferential statistics techniques, such as constructing confidence intervals and conducting hypothesis tests. Since these methods often rely on the normality assumption, the CLT allows us to apply these techniques to a wide range of problems, provided the sample size is large enough. 3. Improved accuracy: As the sample size increases, the variance of the sample mean decreases, leading to more accurate estimates of population parameters. The Central Limit Theorem ensures that not only does our estimator become more accurate, but it also follows a known distribution - the normal distribution- making subsequent analysis and interpretation much easier and reliable. 4. Applicability: The Central Limit Theorem is applicable to a broad range of disciplines, including biology, finance, economics, and social sciences. Its wide-ranging applicability makes it a critical tool in understanding and predicting real-world phenomena using large-sample statistical estimation.