Lesson 4: Confidence Intervals

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This lesson corresponds to Chapter 3 in the Lock^5 textbook. In Lessons 2 and 3 you learned about descriptive statistics.  Lesson 4 begins our coverage of inferential statistics.  Inferential statistics use data from a sample to make an inference about a population. 

Lesson 4 Learning Objectives

Upon completion of this lesson, you will be able to:

  • construct and interpret sampling distributions using StatKey.
  • interpret confidence intervals.
  • explain the process of bootstrapping.
  • construct bootstrap confidence intervals using StatKey.
  • construct bootstrap confidence intervals using Minitab Express.

Confidence intervals use data collected from a sample to estimate a population parameter. In this lesson we will be working with the following statistics and parameters:

 Population ParameterSample Statistic
Mean\(\mu\)\(\overline x\)
Standard deviation\(\sigma\)\(s\)
Difference in two means\(\mu_1 - \mu_2\)\(\overline x_1 - \overline x_2\)
Proportion\(p\)\(\widehat p\)
Difference in two proportions\(p_1 - p_2\)\(\widehat p_1 - \widehat p_2\)
Correlation\(\rho\)\(r\)
Slope (simple linear regression)\(\beta\)\(b\)
Parameters are fixed values, but we rarely know them because it is often difficult to obtain measures from the entire population. Statistics are known values computed from a sample; they are random variables because they differ from sample to sample.
image of cars in traffic

Examples

Proportion

A survey is carried out at a university to estimate the proportion of undergraduate students who drive to campus to attend classes. One thousand students are randomly selected and asked whether they drive or not to campus to attend classes. The population is all of the undergraduates at that university. The sample is the group of 1000 undergraduate students surveyed. The parameter is the true proportion of all undergraduate students at that university who drive to campus to attend classes. The statistic is the proportion of the 1000 sampled undergraduates who drive to campus to attend classes.

Mean

A study is conducted to estimate the true mean annual income of all adult residents of California. The study randomly selects 2000 adult residents of California. The population consists of all adult residents of California. The sample is the 2000 residents in the study. The parameter is the true mean annual income of all adult residents of California. The statistic is the mean of the 2000 residents in this sample.

Ultimately, we measure sample statistics and use them to draw conclusions about unknown population parameters. This is statistical inference.