Confidence Intervals with Proportions

Chapter 9

Suppose we wanted to estimate the proportion of the earth that is covered by water?

Point EstimatePoint Estimate

• Use a singlesingle statistic based on sample data to estimate a population parameter

• Simplest approach

• But not always very precise due to variationvariation in the sampling distribution

Confidence intervalsConfidence intervals

• Are used to estimateestimate the unknown population parameter

• Formula:

statistic + margin of error

Margin of errorMargin of error

• Shows how accurateaccurate we believe our estimate is

• The smallersmaller the margin of error, the more precisemore precise our estimate of the true parameter

• Formula:

statistic theof

deviation standard

value

criticalm

Rate your confidenceRate your confidence0 - 1000 - 100

• 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?

What happens to your confidence as the interval gets smaller?

Your confidence level decreases Your confidence level decreases with smaller intervalswith smaller intervals

%%%%

%%%%

Confidence levelConfidence level

• Is the success rate of the methodmethod used to construct the interval

• Using this method, ____% of the time the intervals constructed will containcontain the true population parameter

• Found fromfrom the confidence level• The upper z-scoreupper z-score with probability p lying to

its left under the standard normal curve

Confidence level tail area z*

.05 1.645

.025 1.96

.005 2.576

Critical value (z*)Critical value (z*)

.05

z*=1.645

.025

z*=1.96

.005

z*=2.57690%95%99%

Confidence interval for a Confidence interval for a population proportion:population proportion:

npp 1p̂ *z

Statistic + Critical value × Standard deviation of the statistic

Margin of error

npp ˆ1ˆ

But do we know the population proportion?

What are the steps for performing a confidence interval?1.) Identify the interval by name or

formula (CI for one-sample proportion)2.) Assumptions• SRS of context• Approximate Normal distribution because

np > 10 & n(1-p) > 10• Population is at least 10n

3.) Calculations

4.) Conclusion (in context of problem)

Conclusion Statement:Conclusion Statement: (memorize!!)(memorize!!)

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

Suppose we wanted to estimate the proportion of the earth that is covered by water? Let’s “throw” the earth around and record the number of times that we point to water or land. Repeat the sampling for 50 trials. Calculate a 90% confidence interval for the amount of water on earth.

Calculate a 95% confidence interval for the true proportion of water on the earth.

Calculate a 99% confidence interval for the true proportion of water on the earth.

What do you

notice?

http://www.math.usu.edu/~schneit/CTIS/CI

• As the confidence level increases, do the intervals generally get wider or more narrow? Explain.

wider• As the sample size increases, do the intervals generally get wider or more narrow? Explain.

More narrow•When 100 confidence intervals are generated, why

are they all different? Sampling variability

• If the confidence level selected is 90%, about how many of 100 intervals will cover the true percentage of

orange balls? Will exactly this number of intervals cover the true percentage each time 100 intervals are

created? Explain.

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?

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) = 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

npp

zP

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 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 assumptions!

Step 2: make calculations

Step 3: conclusion in context

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!

npp

zm1

*

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 + 0.04 of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval?

60125.600

25.96.104.

5.5.96.104.

5.5.96.104.

1*

2

n

n

n

n

npp

zm

Use p-hat = .5

Divide by 1.96

Square both sides

Round up on sample size