The following material is from sections 8.1 and 8.2 in the textbook.

Introduction

Be sure to read through all of this and watch the videos at the end!

Consider the following situation. There is a large population of data that you are interested in. In particular, you would like to know the mean of this population. Since the population is so large, you can't calculate this directly, so you will estimate it. You take a sample of size ~40.

What do you/don't you know?

You don't know the shape of the population.
You don't know the value of the population mean.
You do know the sample mean (you can calculate this from your sample).
You do know that by the CLT, the distribution of sample means is normal.
Therefore, you also know that by the empirical rule, about 95% of sample means will be within two standard errors of the population mean.

This means that if you took your sample mean and added about 2 standard errors and subtracted about 2 standard errors, you would, 95% of the time, have the population mean within that range of values.

This calculation results in what is called a confidence interval. To be more exact, for a 95% confidence interval we will use 1.96 instead of 2. But, the idea is the same.

Two confidence intervals for the mean

There are two approaches for a confidence interval for the mean.

z-interval: This is used when you know the population standard deviation $σ$ .
t-interval: This is used when you do not know the population standard deviation (most common).

Both can be calculated either by hand or using a calculator.

Example videos - Confidence Intervals By Hand

The following videos will help you see how to calculate a z-intervalor a t-interval using the formula.

Z-intervals

t-intervals

Example video - confidence intervals on the calculator

Confidence intervals can also be calculated very quickly using a graphing calculator or other tool like excel. The video below will walk you through using a calculator for this.

You may also find the following links helpful as you study confidence intervals:

Click NEXT to see how to calculate minimum sample sizes.