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The Central Limit Theorem (CLT) is one of the foundational principles of statistics. It states that regardless of the shape of the population distribution (whether skewed, bimodal, or uniform), the sampling distribution of the sample mean will approach a normal distribution as the sample size increases. This phenomenon holds true even if the original population distribution is not normal. However, the convergence to normality improves with larger sample sizes, typically when n≥30n \ geq 30 n≥30, which is often cited as a rule of thumb. For example, consider a population with a skewed distribution, such as household incomes. Individual samples drawn from this population might reflect its skewed nature. However, if you repeatedly take samples and compute their means, plotting these sample means will produce a distribution that becomes increasingly normal as the number of samples grows. This property allows statisticians to apply inferential methods, such as hypothesis testing and confidence intervals, based on the assumption of normality. Why Other Options Are Wrong:
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