# Week 8 or 9 - Normal Distributions and the Central Limit Theorem

**The Central Limit Theorem**

**Why**

The central limit theorem is the most important result in elementary statistics. It will allow us to take any population - skewed, uniform, bi-modal, etc. and convert it to a problem with a normal distribution, provided the samples are random and the sample size is large enough. Once we have a normal distribution there are techniques we can apply to determine anything we want to know about it. This chapter thus serves as our justification for the process used in much of inferential statistics, like confidence intervals and hypothesis tests.

**Learning Objectives**

- Know how to use the normal distribution to solve a problem
- Understand the relationship between probability and the random variable for normal populations
- Understand what a sampling distribution is
- Know the requirements of the central limit theorem
- Know the consequences of the central limit theorem

**Performance Criteria**

- The learner recognizes when the normal distribution is appropriate and translates written problems into statistical notation and/or technology input.
- The learner accurately calculates probabilities and values of the random variable for normal populations using technology.
- The learner draws or sketches normal curves to diagram the relationship between area and probability for normal data.
- The learner will observe the creation of sampling distributions using simulations.
- The learner will verify that requirements for the central limit theorem are met when solving problems.
- The learner will calculate the mean and standard deviation for sampling distributions for the mean.

Click on "Textbook" above to view the reading assignment for this experience or read sections 6.2 and 7.1 from your print or digital copy.

**Helpful Websites!**

Penn State's course on Elementary Statistics

The calculator here will find probabilities and create graphs for Normal Distribution problems. Feel free to use it for generating the graph in the Applications for this experience.

Enter your mean and standard deviation on the right with **Edit Parameters**.

Choose Left Tail, Two-Tail, or Right Tail depending on the problem.

Enter the probability that you are seeking or the x-value(s).

Here is a guide to using StatKey for calculating normal probabilities and using inverse normal calculations.

Syntax for calculating the probabilities and random variables for normal populations

**Videos**

Chapter 6 This is a video by the authors covering chapter 6. (Focus more on the first 22 minutes)

Using Drawings to Solve Normal Distribution Problems

Calculating Probabilities using the Normal Distribution (StatKey)

Calculating Probabilities using the Normal Distribution (Excel)

Calculating Probabilities using the Normal Distribution (TI-84)

**Videos**

Chapter 7 This is a video by the authors covering chapter 7. (Watch the whole video)

Calculating Probabilities for a Sampling Distributions (Excel)

Calculating Probabilities for a Sampling Distributions (TI-84)

Use this site to simulate the creation of sampling distributions.

**Plan**

- Review - Read the above components and post any questions in the forum below.
- Practice - Complete the practice exercises that follow.
- Think - Answer the Critical Thinking questions in the "Critical Thinking" forum.
- Apply - Complete the Application Problems and upload your completed files.

If you have any questions about the content (readings, problems, etc.) then post in the "Questions about Week 8" forum.

**Practice Problem Videos**

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