Module 7: The Central Limit Theorem (Chapter 6)
Section outline
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Why are we so concerned with means? Two reasons are: they give us a middle ground for comparison, and they are easy to calculate. In this chapter, you will study means and the central limit theorem.
The central limit theorem (clt for short) is one of the most powerful and useful ideas in all of statistics. There are two alternative forms of the theorem, and both alternatives are concerned with drawing finite samples size n from a population with a known mean, μ, and a known standard deviation, σ.
The first alternative says that if we collect samples of size n with a "large enough n," calculate each sample's mean, and create a histogram of those means, then the resulting histogram will tend to have an approximate normal bell shape. The second alternative says that if we again collect samples of size n that are "large enough," calculate the sum of each sample and create a histogram, then the resulting histogram will again tend to have a normal bell-shape.
In either case, it does not matter what the distribution of the original population is, or whether you even need to know it. The important fact is that the distribution of sample means and the sums tend to follow the normal distribution.
The size of the sample, n, that is required in order to be "large enough" depends on the original population from which the samples are drawn (the sample size should be at least 30 or the data should come from a normal distribution). If the original population is far from normal, then more observations are needed for the sample means or sums to be normal. Sampling is done with replacement. Given simple random samples of size n from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution.
It would be difficult to overstate the importance of the central limit theorem in statistical theory. Knowing that data, even if its distribution is not normal, behaves in a predictable way is a powerful tool.
Image Caption: If you want to figure out the distribution of the change people carry in their pockets, using the central limit theorem and assuming your sample is large enough, you will find that the distribution is normal and bell-shaped. (credit: John Lodder)
(Content & Image Source: Chapter 7 Introduction, Introductory Statistics, Barbara Illowsky and Susan Dean, OpenStax, CC BY 4.0 License)
Upon completion of this module, you will be able to:6.3 The Central Limit Theorem for Sample Means (Averages)- Recognize characteristics of the Central Limit Theorem for the Sample Means
- Apply and interpret the the Central Limit Theorem for the Sample Means and use to solve real-world applications.
6.4 The Central Limit Theorem for Sums- Recognize characteristics of the Central Limit Theorem for the Sums
- Apply and interpret the the Central Limit Theorem for the Sums and use to solve real-world applications.
6.5 The Normal Approximation to the Binomial- Approximate the binomial distribution using the normal distribution
To achieve these objectives:- Read the Module 7 Introduction (see above).
- Read Sections 6.3 - 6.4 of Chapter 6: The Central Limit Theorem in Introductory Statistics (links to each Section provided below)
- Complete the MyOpenMath Homework Assignments for the topics in the Chapter (links provided below) - These are graded!
- View the Chapter 6 Review (link provided below)
- Practice the problems in the Chapter 6 Practice and Homework, checking the solutions provided (links to each provided below)
- Submit the Chapter 6 Project III: Pocket Change or the Chapter 6 Project IV: Cookie Recipes (links to project and submission link provided below)
- Complete the MyOpenMath Quiz for Chapter 6 (CLT) (link provided below) - This is graded!
- Once you complete the Quiz, upload your work in the Quiz Work Upload Assignment using the submission link below.
- Post in the Chapter 6 Q&A Discussion Forum (CLT) - link provided below.
Note the check boxes to the right that help you track your progress: some are automatic, and some are manual.Module Pressbooks Resources and Activities
You will find the following resources and activities in this module at the Pressbooks website. Click on the links below to access or complete each item.
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