### Unit 4: Estimation with Confidence Intervals

In this unit, you will learn how to use the central limit theorem and confidence intervals, the latter of which enables you to estimate unknown population parameters. The central limit theorem provides us with a way to make inferences from samples of non-normal populations. This theorem states that given any population, as the sample size increases, the sampling distribution of the means approaches a normal distribution. This powerful theorem allows us to assume that given a large enough sample, the sampling distribution will be normally distributed.

You will also learn about confidence intervals, which provide you 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 point estimates will vary from sample to sample, so an interval with certain confidence level is better than a single point estimate. After completing this unit, you will know how to construct such confidence intervals and the level of confidence.

**Completing this unit should take you approximately 10 hours.**

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

- explain the central limit theorem, and use it to construct confidence intervals;
- compare t-distributions and normal distributions;
- apply and interpret the central limit theorem for sample averages;
- calculate, describe, and interpret confidence intervals for population averages and one population proportions; and
- interpret the student-t probability distribution as the sample size changes.

### 4.1: Point Estimators and Their Characteristics

### 4.1.1: Sample Statistics and Parameters

Read sections 2 and 3 from Chapter 10. Also, complete the questions in each section. Section 2 explains the basic concepts of sample statistics and population parameters as well as the basic goal of estimation for which point estimates and interval estimates are introduced. Section 3 talks about the degree of freedom, which is defined as the number of independent pieces of information on which a point estimate is based. Section 3 also talks about variance, a quantity depending on the degrees of freedom.

### 4.1.2: Bias and Sampling Variability

Read section 4 from Chapter 10. Also, complete the questions at the end. Section 4 discusses two important characteristics used as point estimates of parameters: bias and sampling variability. Bias refers to whether an estimator tends to over or underestimate the parameter. Sampling variability refers to how much the estimate varies from sample to sample.

### 4.2: Confidence Intervals

### 4.2.1: Confidence Intervals for Mean

Read sections 7, 8, 9, and 11 from Chapter 10. Also, answer the questions at the end of each section. Section 7 explains the need for confidence intervals and why a confidence interval is not the probability the interval contains the parameter. Section 8 explains how to compute a confidence interval on the mean when sigma is unknown and needs to be estimated. For this purpose, it also explains when to use t-distribution or a normal distribution. Section 9 states the difference between the shape of the t distribution and the normal distribution, and this section also explains how this difference is affected by degrees of freedom. Section 11 explains the procedure to compute a confidence interval on the difference between means.

This lab will help you develop a basic understanding of the properties of a sampling distribution, based on the properties of the population. Review the illustrated instructions, and follow the general instructions to learn more about confidence interval simulations.

Then, use the Wolfram Demonstration provided in the next resource for a more interactive practice of the topics covered here.

Note: Running the actual

*Online Statistics Education*simulation yourself will likely not work, as the Java format that it is is no longer supported by most browsers.The demonstration provided here is a supplement to the reading above. Use the information provided on the demonstration page and interact with the simulation. If you prefer to watch a video of this simulation, you may do so here.

Note: If you have not done so already, you will need to download install the free CDF Player from the Wolfram Demonstrations Project. If using Chrome as your browser, you will also need to download the CDF file from the page linked to above, and run it through the CDF Player on your desktop. Other browsers will allow you to interact with the demonstration directly on the webpage.

Read the instructions and watch the video demo in order to see how the degrees of freedom affect the difference between t and normal distributions.

Note: Running the actual simulation yourself will likely not work, as the Java format that it is in is no longer supported by most browsers. Instead, you may use the Wolfram Demonstrations provided below for a more interactive practice of the topics covered here.

The demonstration provided here are a supplement to the readings and videos above. Use the information provided on the demonstration page and interact with the simulation. If you prefer to watch a video of this simulation, you may do so here.

Note: If you have not done so already, you will need to download install the free CDF Player from the Wolfram Demonstrations Project. If using Chrome as your browser, you will also need to download the CDF file from the page linked to above, and run it through the CDF Player on your desktop. Other browsers will allow you to interact with the demonstration directly on the webpage.

### 4.2.2: Confidence Intervals for Correlation and Proportion

Read sections 12 and 13 from Chapter 10. Also, complete the questions at the end of each section. Section 12 shows how to compute a confidence interval for Pearson's correlation; the solution lies in using Fisher's z transformation. Section 13 explains the procedure to compute confidence intervals for population proportions, where the sampling distribution needs a normal approximation.

Watch these two videos, which discuss confidence intervals.

### Unit 4 Assessment

Take this assessment to see how well you understood this unit.

- This assessment
**does not count towards your grade**. It is just for practice! - You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.

- This assessment