### 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 4 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

First, we'll discuss the basic concepts of sample statistics and population parameters. Then, we'll talk about the degree of freedom, which is the number of independent pieces of information that a point estimate is based on. Finally, we will talk about variance, which depends on the degrees of freedom.

### 4.1.2: Bias and Sampling Variability

This section 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

This section explains the need for confidence intervals and why a confidence interval is not the probability the interval contains the parameter. Then, it discusses how to compute a confidence interval on the mean when sigma is unknown and needs to be estimated. It also explains when to use t-distribution or a normal distribution. Next, it covers the difference between the shape of the t distribution and the normal distribution and how this difference is affected by degrees of freedom. Finally, it explains the procedure to compute a confidence interval on the difference between means.

This demonstration provided here is a supplement to the previous section.

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

This demonstration is a supplement to the previous materials.

### 4.2.2: Confidence Intervals for Correlation and Proportion

- First, this section shows how to compute a confidence interval for Pearson's correlation. The solution uses Fisher's z transformation. Then, it explains the procedure to compute confidence intervals for population proportions where the sampling distribution needs a normal approximation.
Watch these videos, which discuss confidence intervals.

### Unit 4 Assessment

- Receive a grade
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