AP Statistics: Central Limit Theorem

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Topic: Central Limit Theorem

The Central Limit Theorem is a fundamental concept in statistics that states that the distribution of sample means from any population will approximate a normal distribution as the sample size increases, regardless of the shape of the original population distribution. This theorem is crucial in statistical inference and hypothesis testing.

According to the Central Limit Theorem, if we take repeated random samples of size n from a population with a mean μ and a standard deviation σ, the distribution of the sample means will be approximately normal, with a mean equal to the population mean μ and a standard deviation equal to the population standard deviation divided by the square root of the sample size .

This theorem is particularly useful when working with large sample sizes, as it allows us to make inferences about the population parameters based on the sample statistics. It also enables us to perform hypothesis tests and construct confidence intervals.

Example

To better understand the Central Limit Theorem, consider an example where we are interested in studying the average height of a population. We collect random samples of different sizes from the population and calculate the sample means. As the sample size increases, the distribution of these sample means will become more and more normally distributed, regardless of the original population distribution.

Exercise

To practice applying the Central Limit Theorem, consider the following exercise:

A company wants to determine the average time it takes for their customer service representatives to resolve a customer issue. They collect a random sample of 100 customer service interactions and calculate the average resolution time. Based on the Central Limit Theorem, explain why the distribution of these sample means is likely to be approximately normal.

Resources

For further study on the Central Limit Theorem, refer to the following resources:

Video: Central Limit Theorem Explained

Article: Central Limit Theorem