Inference on Proportions

Rate your confidence 0 - 100 Guess my age within 10 years? 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 smaller the interval, the lower your confidence. % % % %

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

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

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%

What does it mean to be 95% confident? 95% chance that  is contained in the confidence interval The probability that the interval contains  is 95% The method used to construct the interval will produce intervals that contain  95% of the time.

Statement: (memorize!!) We are ________% confident that the true proportion context lies within the interval ______ and ______.

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

Assumptions: SRS of context Approximate Normal distribution because np > 10 & n(1-p) > 10 Population is at least 10n

Formula for Confidence interval: Normal curve Note: For confidence intervals, we DO NOT know p – so we MUST substitute p-hat for p in both the SD & when checking assumptions.

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%.

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!

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

How can you make the margin of error smaller? z* smaller (lower confidence level) s smaller (less variation in the population) n larger (to cut the margin of error in half, n must be 4 times as big) Really cannot change!

Hypothesis Tests

Hypotheses for proportions: H0: p = value Ha: p > value where p is the true proportion of context Use >, <, or ≠

Formula for hypothesis test:

A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company’s product. The radio station conducts a random sample of 400 people and finds that 90 have heard the ad and recognize the product. Is this sufficient evidence for the company to renew its contract?

Use the parameter in the null hypothesis to check assumptions! Have an SRS of people np = 400(.2) = 80 & n(1-p) = 400(.8) = 320 - Since both are greater than 10, this distribution is approximately normal. Population of people is at least 4000. Use the parameter in the null hypothesis to check assumptions! H0: p = .2 where p is the true proportion of people who Ha: p > .2 heard the ad Use the parameter in the null hypothesis to calculate standard deviation! Since the p-value >a, I fail to reject the null hypothesis. There is not sufficient evidence to suggest that the true proportion of people who heard the ad is greater than .2.