Confidence Intervals for Proportions Chapter 19 Confidence Intervals for Proportions

Topics Confidence Intervals Margin of Error Interpretation

Confidence Intervals Is it reasonable to expect that a population proportion will be exactly the same as a randomly selected sample proportion? Using a sample proportion as an estimate for a population proportion is called using a point estimate.

Confidence Interval A confidence interval gives a range of possible values the proportion lies in.

Confidence Interval Example A random sample of 540 votes indicates that 57% of the sample has voted for a particular candidate in a 2 person race. Is this a large enough lead to call the election?

Assumptions Independence Randomization 10% Condition Sample Size Assumption

Graph of the Problem

Confidence Interval Procedure Establish the confidence level. Find the standard error. Find the interval: Interpret the Confidence Interval.

Example ctd Confidence Level: Standard Error: Interval: Interpretation:

Confidence Levels and Critical Values 90% Confidence Level → z*= 1.645 95% Confidence Level → z*= 1.96 99% Confidence Level → z*= 2.576

Example 80 out of 180 respondents surveyed are in favor of a new city tax on water usage during the summer. Find a 90% CI Find a 95% CI Find a 99% CI

Interpretations Suppose we have a survey that tells us that 68% of college sophomores have already fulfilled their math requirement with a margin of error of 3.2%. The margin of error is obtained at the 90% confidence level. Which of the following interpretations are correct?

Math Requirement ctd. 68% of college sophomores have completed the math requirement. It is probably true that 68% of college sophomores have completed the math requirement. We know that between 64.8% and 71.2% of college sophomores have completed the math requirement. We don’t know what proportion of college sophomores have completed the math requirement, but the interval 64.8% to 71.2% probably contains the true proportion. We are 90% confident that between 64.8% and 71.2% of college sophomores have completed the math requirement.

Confidence and Precision When the sample size remains fixed, increasing the confidence level results in widening the confidence interval (or decreasing the precision.) Conversely, decreasing the confidence level results in narrowing the confidence interval (or increasing the precision.)

Margin of Error For a fixed confidence level, the margin of error (or maximum error of the estimate) is half the length of the confidence interval.

Example What sample size needs to be surveyed to find a margin of error on a close ballot measure to obtain a margin of error within 2% at the 99% confidence level?

Example An election is being held in a county regarding an upcoming school levy vote. Volunteers want to attempt to accurately predict the election results coming up. The levy needs a 65% approval to pass. A sample of 72 voters is taken in which 55 say they will vote in favor of the new levy. Should the school district be confident the levy will pass?

Final Note on Terminology If all conditions and assumptions are met so that a confidence interval for a proportion can be created, this is often known as a one-proportion z-interval.