This is the main navigation page for Unit 4 of the course Introduction to Statistical Analysis, developed using openly licensed materials from Saylor.org's Introduction to Statistics. Below you will find a full description of Unit 4 in general, as well as for each subunit. Follow the links within each subunit description to access particular topics, or proceed directly to the Unit 4 Content Page.

## UNIT 4: CENTRAL LIMIT THEOREM AND CONFIDENCE INTERVALS

In this unit, you will learn how to use the central limit theorem and confidence intervals, the latter of which enables us to estimate unknown population parameters. The central limit theorem provides us with a way to make inference from samples of non-normal populations. This theorem states that given any population (and it does not matter whether or not it is a normal distribution), as the sample size increases, the sampling distribution of the means approaches a normal distribution. It is a powerful theorem because it allows us to assume that given a large enough sample, the sampling distribution will be normally distributed. The central limit theorem is one of the most important ideas in statistics, so be sure to spend time on it.

In this unit, you will also learn about confidence intervals, which provide us with a way to estimate a population parameter. Instead of giving just a one-number estimate of a variable, a confidence interval gives a range of likely values for it. This is useful because sample results will vary from sample to sample, so a range of values is better than a one-number estimate. After completing this unit, you will know how to construct confidence intervals and calculate the margin of error for them. You will learn to how to come up with a range of values for a parameter and the level of confidence for the intervals.

For example, suppose you want to know the amount of soda that an average high school student in New York drinks per day. The average volume of soda for the entire population of New York high school students who drink soda is the parameter you are trying to estimate. Suppose you take a random sample and find out the average amount is 0.5 litres. Then you also want to know how much you expect the average to vary from one sample to the next, with a certain level of confidence. The number that you use to represent this precision, i.e. to measure how close you expect your results to be to the truth, is called the margin of error.

Time Advisory: This unit will take you 11 hours to complete.

### Learning Outcomes

Upon completion of this this unit, you will be able to:

• Recognize the Central Limit Theorem problems.
• Classify continuous word problems by their distributions.
• Apply and interpret the Central Limit Theorem for Averages.
• Apply and interpret the Central Limit Theorem for Sums.
• Calculate and interpret confidence intervals for one population average and one population proportion.
• Interpret the student-t probability distribution as the sample size changes.
• Discriminate between problems applying the normal and the student-t distributions.

### Subunits

Unit four consists of two main topics:

## About the Resources in This Course

This course project draws upon three main types of resources:

The first are readings and video lectures from Barbara Illowsky and Susan Dean’s Collaborative Statistics, which is available freely under a Creative Commons Attribution 2.0 Generic (CC BY 2.0) license from the following location: http://cnx.org/content/col10522/latest/

The second type of resources in this course are lectures from Kahn Academy. These lectures are available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported (CC BY-NC-SA 3.0) license. Kahn Academy has many lectures available from http://www.khanacademy.org/

Finally, the above resources have been woven together and organized into a format analogous to a traditional college-level course by professional consultants that work as experts within the subject area. This process was facilitated by The Saylor Foundation. Additionally, if you have worked through all of the material contained in this project, you may be interested in taking the final exam provided by Saylor.org or completing other courses available there that are not yet on Wikiversity.