Central Limit Theorem - Practice problem

The Central Limit Theorem (CLT) is a fundamental principle in statistics that states that the distribution of sample means will tend to be normally distributed, regardless of the shape of the population distribution, as long as the sample size is sufficiently large. This theorem holds true under certain conditions, notably: Independence: The samples must be independent. This means that the selection of one sample does not influence the selection of another. Sample Size: The sample size should be sufficiently large. While there is no specific number that defines “large,“ a common rule of thumb is that a sample size of 30 or more is often adequate. Identically Distributed Variables: The data must come from the same distribution, but this distribution does not need to be normal. The central limit theorem is crucial because it allows for various statistical methods to be applied, including hypothesis testing and the construction of confidence intervals, even when the underlying population distribution is not normal. It’s particularly useful because it enables statisticians to make inferences about population parameters based on sample statistics. The Central Limit Theorem (CLT) does not apply to a single observation or a very small sample from a population; rather, it pertains to the distribution of sample means over many samples, particularly as the sample size becomes large. To clarify, the CLT states that if you take sufficiently large random samples from a population (with replacement) and calculate the mean of each sample, the distribution of these sample means will approximate a normal distribution, regardless of the shape of the population distribution. This approximation improves as the sample size increases. So, when considering a single sample or a single data point, the CLT does not apply. The theorem is about the behavior of averages of samples as the number of samples increases, and specifically, it’s about the distribution of these averages, not about the distribution of individual data points or a single sample. Problem: If a random sample of 1,000 discharges were taken from the California all-discharge database, and a histogram were made of patient length of stay for the sample, which of the following is most likely true: A) The histogram will look approximately like a normal distribution because the sample size is large , and the Central Limit Theorem applies. B) The histogram will look approximately like a normal distribution because the number of samples is large , and the Central Limit Theorem applies. C) The histogram will appear to be right skewed. D) The histogram will appear to be left skewed. E) The histogram will look like a uniform distribution