Module 8: Confidence Intervals
Introduction: Confidence Intervals
Barbara Illowsky & OpenStax et al.
Learning Objectives
- Calculate and interpret confidence intervals for estimating a population mean and a population proportion.
- Interpret the Student’s t probability distribution as the sample size changes.
- Discriminate between problems applying the normal and the Student’s t distributions.
- Calculate the sample size required to estimate a population mean and a population proportion given a desired confidence level and margin of error.
Suppose you were trying to determine the mean rent of a two-bedroom apartment in your town. You might look in the classified section of the newspaper, write down several rents listed, and average them together. You would have obtained a point estimate of the true mean. If you are trying to determine the percentage of times you make a basket when shooting a basketball, you might count the number of shots you make and divide that by the number of shots you attempted. In this case, you would have obtained a point estimate for the true proportion.
We use sample data to make generalizations about an unknown population. This part of statistics is called inferential statistics. The sample data help us to make an estimate of a population parameter. We realize that the point estimate is most likely not the exact value of the population parameter, but close to it. After calculating point estimates, we construct interval estimates, called confidence intervals.
In this chapter, you will learn to construct and interpret confidence intervals. You will also learn a new distribution, the Student’s-t, and how it is used with these intervals. Throughout the chapter, it is important to keep in mind that the confidence interval is a random variable. It is the population parameter that is fixed.
If you worked in the marketing department of an entertainment company, you might be interested in the mean number of songs a consumer downloads a month from iTunes. If so, you could conduct a survey and calculate the sample mean, [latex]displaystyleoverline{x}[/latex], and the sample standard deviation, s. You would use [latex]displaystyleoverline{x}[/latex] to estimate the population mean and s to estimate the population standard deviation. The sample mean, [latex]displaystyleoverline{x}[/latex], is the point estimate for the population mean, μ. The sample standard deviation, s, is the point estimate for the population standard deviation, σ.
Each of [latex]displaystyleoverline{x}[/latex] and s is called a statistic.
A confidence interval is another type of estimate but, instead of being just one number, it is an interval of numbers. The interval of numbers is a range of values calculated from a given set of sample data. The confidence interval is likely to include an unknown population parameter.
Suppose, for the iTunes example, we do not know the population mean μ, but we do know that the population standard deviation is σ = 1 and our sa