Confidence Intervals with Proportions

1. Confidence Intervals with Proportions Chapter 19

2. Suppose we wanted to estimate the proportion of pennies in this jar of change. How might we go about estimating this proportion?

3. 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

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

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

6. 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?

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

8. 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

9. .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%

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

11. Suppose we wanted to estimate the number proportion of pennies in this jar of change. Let’s take a sample of 20 coins and create a 90% confidence interval.

12. Calculate a 95% confidence interval for the true proportion of pennies in the jar. Calculate a 99% confidence interval for the true proportion of pennies in the jar. What do you notice?

13. Chip Activity What do you notice about these confidence intervals?

14. Blue Chip Activity - 90% Confidence Intervals A set of 50 confidence intervals are displayed below: 0.05 0.90 0.05 P - 1.645 std dev P + 1.645 std dev • How many confidence intervals would contain the true population proportion if confidence intervals were created for ALL possible sample proportions? • The blue confidence intervals were created from sample proportions (p-hats) from the “middle or central” 90% of the sampling distribution. • The red confidence intervals were created from sample proportions (p-hats) from either the lower or upper 5% of the sampling distribution. • Since, in the long run, you will get p-hats from the central region about 90% of the time, for a given set of confidence intervals approximately 90% of them will contain the true population proportion. 90% of them would contain the true population proportion P

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

16. Assumptions: 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

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

18. 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.

19. Assumptions: • 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 assumptions! 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%.

20. 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.

21. Assumptions: • 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 assumptions! 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%.

22. To find sample size: However, since we have not yet taken a sample, we do not know a p-hat (or p) to use! Another Gallop Poll is taken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within + 0.04 of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval?

23. .1(.9) = .09 .2(.8) = .16 .3(.7) = .21 .4(.6) = .24 .5(.5) = .25 By using .5 for p-hat, we are using the worst-case scenario and using the largest SD in our calculations. What p-hat (p) do you use when trying to find the sample size for a given margin of error? Remember that, in a binomial distribution, the histogram with the largest standard deviation was the one for probability of success of 0.5.

24. Another Gallop Poll is taken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within + 0.04 of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval? Use p-hat = .5 Divide by 1.96 Square both sides Round up on sample size