Inference on Proportions

Assumptions:

• SRS

• Normal distribution

np > 10 & n(1-p) > 10

• Population is at least 10n

Formula for Confidence interval:

statistic of SD valuecritical statisticCI

p̂

npp 1*z

Normal curve

Note: For confidence intervals, we DO NOT know p – so we MUST substitute p-hat for pin 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.

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 ghost is between 35% and 41%.

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.

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

Stop & do homework!

Hypotheses for proportions:

H0: p = value

Ha: p > value

where p is the true proportion of context

Use >, <, or ≠

Formula for hypothesis test:

statistic of SD

parameter - statisticstatisticTest

z npp

pp

1

ˆ

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?

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.

H0: p = .2 where p is the true proportion of people who

Ha: p > .2 heard the ad

05.α1056.25.1

400)8(.2.

2.225.

valuepz

Since the p-value >, 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.

Use the parameter in the null hypothesis to check assumptions!

Use the parameter in the null hypothesis to calculate standard

deviation!