Confidence Intervals with Proportions Chapter 9 Notes: Page 165

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

Point Estimate singleUse a single statistic based on sample data to estimate a population parameter Simplest approach variationBut not always very precise due to variation in the sampling distribution

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

Margin of error accurateShows how accurate we believe our estimate is smaller more preciseThe smaller the margin of error, the more precise our estimate of the true parameter Formula:

Rate your confidence Guess my age within 10 years? within 5 years? within 1 year? Shooting a basketball at a wading pool, will you make the basket? Shooting the ball at a large trash can, will you make the basket? Shooting the ball at a carnival, will you make the basket?

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

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

fromFound from the confidence level upper z-scoreThe upper z-score with probability p lying to its right under the standard normal curve Confidence leveltail areaz* Critical value (z*).05 z*= z*= z*= % 95% 99%

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

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. Penny, Nickel, Dime, Quarter P(penny) = ¼ or.25 = [.09,.41]

Calculate a 95% confidence interval for the true proportion of pennies in the jar. = [.06,.44] Calculate a 99% confidence interval for the true proportion of pennies in the jar. = [0,.5] What do you notice? The higher your confidence, the larger the interval. the larger the interval.

Blue Chip Activity What do you notice about these confidence intervals?

P 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. Blue Chip Activity - 90% Confidence Intervals 0.05 P std devP std dev 0.90 A set of 50 confidence intervals are displayed below: How many confidence intervals would contain the true population proportion if confidence intervals were created for ALL possible sample proportions? 90% of them would contain the true population proportion

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

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

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.

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 We are 90% confident that the true proportion of egg cartons with at least one broken egg is between 12.2% and 19.8%. Step 1: check assumptions! Step 2: make calculations Step 3: conclusion in context

Another Gallop Poll istaken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within 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!

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. Remember that, in a binomial distribution, the histogram with the largest standard deviation was the one for probability of success of 0.5.

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 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

Homework: Page 167