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The Central Limit Theorem (CLT) states that, given a large enough sample size (n) from a population with any arbitrary distribution and a finite mean (μ) and standard deviation (σ), the sampling distribution of the sample mean will approach a normal distribution with a mean equal to the population mean (μ) and a standard deviation equal to the population standard deviation (σ) divided by the square root of the sample size. Mathematically, this can be represented as: $$\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)$$ where \(\bar{X}\) is the sample mean, and \(N\left(\mu, \frac{\sigma^2}{n}\right)\) represents a normal distribution with mean μ and variance \(\frac{\sigma^2}{n}\).

The Central Limit Theorem holds great value in large-sample statistical estimation because: 1. It provides a basis for making inferences about the population mean, even if the original population does not follow a normal distribution. This is particularly useful when the true population distribution is unknown or difficult to work with. 2. With the CLT, we can apply the properties of the normal distribution to our sample mean when making inferences about the population mean. This enables us to calculate confidence intervals, hypothesis tests, and other statistical measures that rely on the normal distribution. 3. The CLT simplifies statistical modeling and analysis since the normal distribution has several well-known properties and characteristics that are easy for researchers to work with. 4. It helps to justify using the mean as a measure of central tendency for various populations since, under large-sample conditions, the sample mean follows a normal distribution regardless of the shape of the original distribution. In summary, the Central Limit Theorem is a powerful and essential tool in large-sample statistical estimation, allowing for the use of normal distribution assumptions when working with different population distributions.