The \( \qquad \) tells us that for a population with any distribution, the distribution of the sample means approaches a normal distribution as the sample size increases. The \( \square \) tells us that for a population with any distribution, the distri \( \square \) yution as the sample size increases. Central Limit Theorem 5\% Guideline for Cumbersome Calculations Rare Event Rule Rule Range of Thumb

The Central Limit Theorem (CLT) is a cornerstone of statistics that states that, no matter the shape of the population distribution, the distribution of sample means will tend to be normally distributed if you take sufficiently large samples (usually n ≥ 30). This magical property allows statisticians to make inferences about population parameters even when the population itself is skewed or not normally distributed. In real-world applications, the CLT is crucial for conducting hypothesis tests and constructing confidence intervals. It means that we can confidently use the normal distribution to analyze sample data and draw conclusions about the entire population, making it a powerful tool in fields ranging from medicine to marketing, where decision-making relies on understanding variables from sampled data.