Week 8Confidence Intervals

Unit Overview

Last week we learned a critical new concept, the "standard error" of a sample estimate like a mean or proportion. Now we get to apply that concept to generate something you are already somewhat familiar with: the confidence interval. Confidence intervals have three parts: (1) the degree of confidence you want = 90, 95, 99%, (2) a point estimate = sample mean or proportion, and (3) an error boundary = plus/minus some amount of uncertainty.

Here is an illustration: According to a 2017 Gallup Poll ("telephone interviews conducted July 20-Oct. 19, 2017, on the Gallup U.S. Daily survey, with a random sample of 46,663 adults, aged 18 and older, living in all 50 U.S. states and the District of Columbia"), President Trump's approval rating was 37% ±1 percentage point, measured at a 95% confidence level. See the three parts? Confidence level = 95%, point estimate = .37 or 37 percent, and the error bound is plus or minus .01 or 1 percentage point.

We also will learn another probability distribution -- we use the t-distribution instead of the standard normal (z) distribution in real life, and its shape depends on how big a sample (n) you have. Specifically, its shape is defined by something called its "degrees of freedom" which is equal to n-1 in this instance.

The key concepts this week are:

Confidence interval for a sample mean and a sample proportion
Point estimate
Level of confidence
Student's t-distribution
Degrees of freedom in the t-distribution (n-1)

Unit Learning Objectives

At the completion of this unit/week/section, students will be able to:

Calculate 95% and 99% confidence intervals for a sample mean and for a sample proportion.
Interpret what a confidence interval is -- what exactly are we 95% confident in?
Differentiate between a point estimate and the error bound around that estimate.
Draw a confidence interval on a number line: lowest possible value, highest, & point estimate.
Switch from using the normal distribution to the t-distribution to determine how many standard errors from the point estimate we have to go to encompass 95% or 99% of all possible estimates.

Unit Materials

For this unit, you must read and complete practice problems on the following material:

The Introduction and Sections 8.1, 8.2, and 8.3 of Chap 8 in Illowsky & Dean (2016). Introductory Statistics. OpenStax, ISBN 1938168208, www.openstax.org/details/introductory-statistics.

Assignments & Due dates