1. Section 6.3 Confidence Intervals for Population Proportions © 2012 Pearson Education, Inc. All rights reserved. 1 of 83

2. Section 6.3 Objectives Find a point estimate for the population proportion Construct a confidence interval for a population proportion Determine the minimum sample size required when estimating a population proportion © 2012 Pearson Education, Inc. All rights reserved. 2 of 83

3. Point Estimate for Population p Population Proportion The probability of success in a single trial of a binomial experiment. Denoted by p Point Estimate for p The proportion of successes in a sample. Denoted by – – read as “p hat” © 2012 Pearson Education, Inc. All rights reserved. 3 of 83

4. Point Estimate for Population p Point Estimate for q, the proportion of failures Denoted by Read as “q hat” Estimate Population Parameter… with Sample Statistic Proportion: p © 2012 Pearson Education, Inc. All rights reserved. 4 of 83

5. Example: Point Estimate for p In a survey of 1000 U.S. adults, 662 said that it is acceptable to check personal e-mail while at work. Find a point estimate for the population proportion of U.S. adults who say it is acceptable to check personal e-mail while at work. (Adapted from Liberty Mutual) Solution: n = 1000 and x = 662 © 2012 Pearson Education, Inc. All rights reserved. 5 of 83

6. p. 327 Try it yourself 1

7. Confidence Intervals for p A c-confidence interval for the population proportion p The probability that the confidence interval contains p is c. © 2012 Pearson Education, Inc. All rights reserved. 7 of 83

8. Constructing Confidence Intervals for p 1.Identify the sample statistics n and x. 2.Find the point estimate 3.Verify that the sampling distribution of can be approximated by the normal distribution. 4.Find the critical value z c that corresponds to the given level of confidence c. Use the Standard Normal Table In WordsIn Symbols © 2012 Pearson Education, Inc. All rights reserved. 8 of 83

9. Constructing Confidence Intervals for p 5.Find the margin of error E. 6.Find the left and right endpoints and form the confidence interval. Left endpoint: Right endpoint: Interval: In WordsIn Symbols © 2012 Pearson Education, Inc. All rights reserved. 9 of 83

10. Example: Confidence Interval for p In a survey of 1000 U.S. adults, 662 said that it is acceptable to check personal e-mail while at work. Construct a 95% confidence interval for the population proportion of adults in the U.S. adults who say that it is acceptable to check personal e- mail while at work. Solution: Recall © 2012 Pearson Education, Inc. All rights reserved. 10 of 83

11. Solution: Confidence Interval for p Verify the sampling distribution of can be approximated by the normal distribution Margin of error: © 2012 Pearson Education, Inc. All rights reserved. 11 of 83

12. Solution: Confidence Interval for p Confidence interval: Left Endpoint:Right Endpoint: © 2012 Pearson Education, Inc. All rights reserved. 12 of 83

13. Solution: Confidence Interval for p With 95% confidence, you can say that the population proportion of U.S. adults who say that it is acceptable to check personal e-mail while at work is between ____% and ____%. Point estimate © 2012 Pearson Education, Inc. All rights reserved. 13 of 83

14. p. 329 Try it yourself

15. Sample Size Given a c-confidence level and a margin of error E, the minimum sample size n needed to estimate p is This formula assumes you have an estimate for and. If not, use and © 2012 Pearson Education, Inc. All rights reserved. 15 of 83

16. Example: Sample Size You are running a political campaign and wish to estimate, with 95% confidence, the proportion of registered voters who will vote for your candidate. Your estimate must be accurate within 3% of the true population. Find the minimum sample size needed if 1.no preliminary estimate is available. Solution: Because you do not have a preliminary estimate for use and © 2012 Pearson Education, Inc. All rights reserved. 16 of 83

17. Solution: Sample Size c = 0.95 z c = 1.96 E = 0.03 Round up to the nearest whole number. With no preliminary estimate, the minimum sample size should be at least _______ voters. © 2012 Pearson Education, Inc. All rights reserved. 17 of 83

18. Example: Sample Size You are running a political campaign and wish to estimate, with 95% confidence, the proportion of registered voters who will vote for your candidate. Your estimate must be accurate within 3% of the true population. Find the minimum sample size needed if 2.a preliminary estimate gives. Solution: Use the preliminary estimate © 2012 Pearson Education, Inc. All rights reserved. 18 of 83

19. Solution: Sample Size c = 0.95 z c = 1.96 E = 0.03 Round up to the nearest whole number. With a preliminary estimate of, the minimum sample size should be at least ______voters. Need a larger sample size if no preliminary estimate is available. © 2012 Pearson Education, Inc. All rights reserved. 19 of 83

20. p. 331 Try it yourself 4

21. Section 6.3 Summary Found a point estimate for the population proportion Constructed a confidence interval for a population proportion Determined the minimum sample size required when estimating a population proportion © 2012 Pearson Education, Inc. All rights reserved. 21 of 83