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? Create a jar with different types of coins . . .

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

statistic + margin of error 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*) z*=1.645 z*=1.96 z*=2.576 .05 .025 .005 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 .05 z*=1.645 .025 .005 z*=1.96 z*=2.576 90% 95% 99%

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

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

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.

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 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%.

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 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 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? To find sample size: However, since we have not yet taken a sample, we do not know a p-hat (or p) to use!

Remember that, in a binomial distribution, the histogram with the largest standard deviation was the one for probability of success of 0.5. What p-hat (p) do you use when trying to find the sample size for a given margin of error? .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.

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.