Confidence Intervals with Proportions

Sea Fan

Suppose we wanted to estimate the proportion of registered voters who are more enthusiastic about voting in this election compared to other years?

Suppose we wanted to estimate the proportion of Dr. Pepper cans that are under-filled?

Use a single statistic based on sample data to estimate a population parameter

Simplest approachBut not always very precise due to variation in the sampling distribution

Point Estimate

Are used to estimate the unknown population parameter

Formula:

statistic + margin of error

Confidence intervals

statistic theofdeviation standard

value

criticalm

Margin of errorShows how accurate we believe our

estimate isThe smaller the margin of error, the

more precise our estimate of the true parameter

Formula:

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?

Rate your confidence0 - 100

What happens to your confidence as the interval gets smaller?

The lower your confidence, the smaller the interval. %

%%

%

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

Confidence level

Found from the confidence levelThe 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

Critical value (z*)

.05

z*=1.645

.025

z*=1.96

.005

z*=2.57690%95%99%

Confidence interval for a population proportion:

npp 1p̂ *z

npp ˆ1ˆ

Statistic + Critical value × Standard deviation of the statistic

Margin of error

But do we know the population proportion?

1. Assumptions

2. Calculations

3. Conclusion

What are the steps for performing a confidence interval?

SRS of contextApproximate Normal distribution becausenp > 10 & n(1-p) > 10

Population is at least 10n

Conditions:Where are the last two assumptions from?

We are ________% confident that the true proportion context is between ______ and ______.

Statement: (memorize!!)

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.

Conditions:•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.

41,.35.1012)62(.38.96.138.1*ˆ

nppzP

We are 95% confident that the true proportion of adults who believe in ghosts is between 35% and 41%.

Step 1: check conditions!

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.

Conditions:•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.

198,.122.250)84(.16.645.116.

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 conditions!

Step 2: make calculations

Step 3: conclusion in context