Question

fill in the blank. the 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 popu means appr 5% guideline for cumbersome calculations rule range of thumb rare event rule central limit theorem

Explanation:

The Central Limit Theorem is a fundamental concept in statistics. It states that regardless of the shape of the population distribution (whether it's uniform, skewed, etc.), as the sample size \(n\) increases, the sampling distribution of the sample mean \(\bar{x}\) approaches a normal distribution. The other options:

• The 5% Guideline for Cumbersome Calculations is about when we can assume independence in sampling without replacement (if \(n\leq 0.05N\), where \(n\) is sample size and \(N\) is population size).
• The Rule of Thumb (in a general sense in stats, like the empirical rule for normal distributions which says about 68 - 95 - 99.7% of data within 1, 2, 3 standard deviations) is not related to the sampling distribution of the mean approaching a normal distribution.
• The Rare Event Rule is used in hypothesis testing (if under a claim the probability of an event is very small, we reject the claim).

Answer:

Central Limit Theorem