Confidence Intervals with Proportions

1. Confidence Intervals with Proportions

2. Sea Fan

3. Suppose we wanted to estimate the proportion of registered voters who are more enthusiastic about voting in this election compared to other years? Suppose we wanted to estimate the proportion of Dr. Pepper cans that are under-filled?

4. Point Estimate • Use a single statistic based on sample data to estimate a population parameter • Simplest approach • But not always very precise due to variation in the sampling distribution

5. Confidence intervals • Are used to estimate the unknown population parameter • Formula: statistic + margin of error

6. Shows how accurate we believe our estimate is The smaller the margin of error, the more precise our estimate of the true parameter Formula: Margin of error

7. Rate your confidence0 - 100 • Guess my age within 10 years? • within 5 years? • within 1 year? • Shooting a basketball at a wading pool, will make basket? • Shooting the ball at a large trash can, will make basket? • Shooting the ball at a carnival, will make basket?

8. What happens to your confidence as the interval gets smaller? The lower your confidence, the smaller the interval. % % % %

9. Confidence level • Is the success rate of the methodused to construct the interval • Using this method, ____% of the time the intervals constructed will contain the true population parameter

10. .05 .025 .005 Critical value (z*) • Found from the confidence level • The upper z-score with probability p lying to its right under the standard normal curve Confidence level tail area z* .05 1.645 .025 1.96 .005 2.576 z*=1.645 z*=1.96 z*=2.576 90% 95% 99%

11. Confidence interval for a population proportion: But do we know the population proportion? Statistic + Critical value × Standard deviation of the statistic Margin of error

12. What are the steps for performing a confidence interval? • Assumptions • Calculations • Conclusion

13. Conditions: Where are the last two assumptions from? • SRS of context • Approximate Normal distribution because np> 10 & n(1-p) > 10 • Population is at least 10n

14. Statement:(memorize!!) We are ________% confident that the true proportion context is between ______ and ______.

15. A May 2000 Gallup Poll found that 38% of a random sample of 1012 adults said that they believe in ghosts. Find a 95% confidence interval for the true proportion of adults who believe in ghost.

16. Conditions: • Have an SRS of adults • np =1012(.38) = 384.56 & n(1-p) = 1012(.62) = 627.44 Since both are greater than 10, the distribution can be approximated by a normal curve • Population of adults is at least 10,120. Step 1: check conditions! Step 2: make calculations Step 3: conclusion in context We are 95% confident that the true proportion of adults who believe in ghosts is between 35% and 41%.

17. The manager of the dairy section of a large supermarket took a random sample of 250 egg cartons and found that 40 cartons had at least one broken egg. Find a 90% confidence interval for the true proportion of egg cartons with at least one broken egg.

18. Conditions: • Have an SRS of egg cartons • np =250(.16) = 40 & n(1-p) = 250(.84) = 210 Since both are greater than 10, the distribution can be approximated by a normal curve • Population of cartons is at least 2500. Step 1: check conditions! Step 2: make calculations Step 3: conclusion in context We are 90% confident that the true proportion of egg cartons with at least one broken egg is between 12.2% and 19.8%.