Confidence Intervals for Proportions

Chapter 19

Rate your confidenceRate your confidence0 - 1000 - 100

• Name 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?

The larger your confidence, the wider the interval.

Confidence intervalsConfidence intervals

• Are used to estimate the unknown population proportion

• Formula:

estimate + margin of error

Margin of errorMargin of error

• Shows how accurate we believe our estimate is

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

• Formula:

statistic theof

deviation standard

value

criticalm

Confidence levelConfidence 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

What does it mean to be 95% What does it mean to be 95% confident?confident?

• 95% chance that p is contained in the confidence interval

• The probability that the interval contains p is 95%

• The method used to construct the interval will produce intervals that contain p 95% of the time.

• 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

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

.05

z*=1.645

.025

z*=1.96

.005

z*=2.57690%95%99%

Formula for Confidence interval:

statistic of SD valuecritical statisticCI

p̂

npp 1*z

Note: For confidence intervals, we DO NOT know p – so we MUST substitute p-hat for pin both the SD & when checking assumptions.

Steps for doing a confidence interval:Steps for doing a confidence interval:

1) State the parameter

2) Assumptions –1)SRS from population

2) Success/Failure Condition (Large enough sample)np and n(1-p)

rule – the sample is less then 10% of the population

3) Calculate the interval

4) Write a statement about the interval in the context of the problem.

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

We are ________% confident that the true proportion context lies within the interval ______ 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 ghosts.

3) The sample should be less than 10% of the population. The population should be at least 10,120 adults which we will assume.

p = the true proportion of adults who believe in ghosts

State the parameter

Justify the confidence interval needed (state assumptions)

1) The sample must be random which is stated in the problem.

2) The sample should be large. Since np =1012(.38) = 384.56 > 10 and n(1-p) = 1012(.62) = 627.44 > 10, the sample is large enough.

Since the conditions are satisfied a CI for proportions is appropriate.

1 .38(.62)ˆ * .38 1.96 .35,.41

1012

p pp z

n

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

Calculate the confidence interval.ˆ .38p

1012n95% CI

Explain the interval in the context of the problem.

If we wanted a 99% confidence interval for the previous problem, what would change?

1 .38(.62)ˆ * .38 2.576 .34,.42

1012

p pp z

n

ˆ .38p 1012n

99% CI

The interval is larger. Only the margin of error changed.

Work page 447 #13

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.

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