1. Section 6.3 Confidence Intervals for Population Proportions Larson/Farber 4th ed

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 Larson/Farber 4th ed

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”

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 Larson/Farber 4th ed

5. Example: Point Estimate for p In a survey of 458 U.S. adults 224 said that they eat meat daily. Find a point estimate for the population proportion of U.S. adults who say they eat meat daily. (Adapted from Greenfield Online) Solution: n = 458 and x = 224

6. Confidence Intervals for p A c-confidence interval for the population proportion p Larson/Farber 4th ed The probability that the confidence interval contains p is c.

7. 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 Larson/Farber 4th ed In WordsIn Symbols

8. Constructing Confidence Intervals for p 5.Find the margin of error E. Find the left and right endpoints and form the confidence interval. Left endpoint: Right endpoint: Interval: Larson/Farber 4th ed In WordsIn Symbols

9. Example: Confidence Interval for p In a survey of 458 U.S. adults 224 said that they eat meat daily. Construct a 95% confidence interval for the proportion of adults in the United States who say that they eat mea daily. Solution: Recall

10. Solution: Confidence Interval for p Verify the sampling distribution of can be approximated by the normal distribution Margin of error:

11. Solution: Confidence Interval for p Confidence interval: Left Endpoint:Right Endpoint: 0.44 < p < 0.54

12. Solution: Confidence Interval for p 0.44 < p < 0.54 With 95% confidence, you can say that the proportion of adults who say they eat meat daily is between 44% and 54%. ( ) 0.490.440.54 Point estimate

13. 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: So what do we do if we do not know the values for

14. Sample Size 0.90.10.09 0.80.20.16 0.70.30.21 0.60.40.24 0.5 0.25 0.40.60.24 0.30.70.21 0.20.80.16 0.10.90.0900000000000 001

15. Example: Sample Size You are a travel agent and wish to estimate, with a 98% confidence, the proportion of vacationers who use an online services to make travel reservations. Your estimate must be accurate within 4% of the true population. Find the minimum sample size needed if no preliminary estimate is available. Solution: Because you do not have a preliminary estimate for use:

16. Solution: Sample Size c = 0.98 z c = 2.33 E = 0.04 Round up to the nearest whole number. With no preliminary estimate, the minimum sample size should be at least 849 voters.

17. Example: Sample Size You are a travel agent and wish to estimate, with a 98% confidence, the proportion of vacationers who use an online services to make travel reservations. Your estimate must be accurate within 4% of the true population. Find the minimum sample size needed if a previous study indicated that 30% of respondents said they used online services to make reservations Solution: Use the preliminary estimate

18. Solution: Sample Size c = 0.98 z c = 2.33 E = 0.04 Round up to the nearest whole number. With a preliminary estimate of 30%, the minimum sample size should be at least 713 voters. Need a larger sample size if no preliminary estimate is available.

19. 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 HW 6.3: 1 - 27 EO