Confidence Intervals with Proportions Chapter 9 Suppose we

Confidence Intervals with Proportions Chapter 9

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

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

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

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:

Rate your confidence 0 - 100 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?

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

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

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 90% 95% 99% z*=1. 645 tail area z*=1. 96 z*=2. 576 z*. 05. 025. 005 1. 645. 025 1. 96. 005 2. 576

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

What are the steps for performing a confidence interval? 1. Assumptions 2. Calculations 3. Conclusion

Assumptions: Where are the last two • assumptions from?

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

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.

Assumptions: Step 1: check 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 Step 2: make calculations • Population of adults is at least 10, 120. Step 3: conclusion in context We are 95% confident that the true proportion of adults who believe in ghosts is between 35% and 41%.

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.

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%.

Review of CI’s for Proportions A recent poll consisted of 1012 randomly selected adults who were asked whether “cloning of humans should or should not be allowed. ” Results showed that 901 of those surveyed said that cloning should not be allowed. Construct a 95% confidence interval of the proportions of adults who believe that cloning of humans should not be allowed. Based on that interval, is there strong evidence to support the claim that the majority is opposed to the cloning of humans? Justify your answer.

Another Gallop Poll is taken To find sample size: in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is However, since we have not yet taken a sample, we do not know a p-hat (or p) to necessary to be within + 0. 04 of the use! true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval?

Remember that, in a binomial What p-hat (p) do you use when distribution, the histogram with the trying to find the sample size for a largest standard deviation was the one given margin of error? for probability of success of 0. 5. . 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 worstcase scenario and using the largest SD in our calculations.

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

Review of CI’s for Proportions You wish to estimate with 90% confidence the proportion of adults 18 to 29 who have high blood pressure. In a previous survey 4% of adults in this age group had high blood pressure. What is the minimum sample size needed if you are to be accurate within 5% of the population proportion?

Review of CI’s for Proportions You are running a political campaign and wish to estimate with 95% confidence, the proportion of registered voters who will vote for your candidate. What is the minimum sample size needed if you are to be accurate within 3% of the population proportion.