Section 12.1

Inference for a Population Proportion

Confidence Intervals

So far, we’ve studied z-procedures Confidence Intervals One sample z-procedures

And t-procedures Confidence Intervals One-sample t-procedures

All of our inference has been for μ, the population mean. What about the population proportion?

Notation Again

Statistic Parameter

Mean

Proportion ˆ

x

p p

Example

How common is behavior that puts people at risk of AIDS? The National AIDS Behavioral Surveys interviewed a SRS of 2673 adult heterosexuals. Of these, 170 had more than one sexual partner in the past year. That’s 6.36% of the sample. Describe the population and explain in words

what the parameter p is. Give the numerical value of the statistic p-hat that

estimates p.

Formulas and Standard Error

x

StandardError

Test StatisticFormula

p̂

n

s

n

pp )ˆ1(ˆ

n

sx

t

npp

ppz

)ˆ1(ˆ

ˆ

Assumptions and Conditions

The data came from an SRS from the population of interest.

The population is at least 10 times the sample size. The population is distributed normally.

Check both: np ≥ 10 and n(1 – p) ≥ 10 We don’t know p!

For confidence intervals, use the sample proportion, p-hat. We will estimate using: n(p-hat) ≥ 10 and n(1 - p-hat) ≥ 10

Note: This is an additional condition not with inference for means.

Carrying Out the Inference

We will follow the same steps for Inference. The changes for proportions:

For confidence intervals, the formula will be:

* ˆ ˆ(1 )ˆCI for p:

p pp z

n

Example

How common is behavior that puts people at risk of AIDS? The National AIDS Behavioral Surveys interviewed a SRS of 2673 adult heterosexuals. Of these, 170 had more than one sexual partner in the past year. That’s 6.36% of the sample. We want to construct a 95% confidence interval for p, the proportion of the population that has had more than one sexual partner in the past year.

Margin of Error

Margin of Error: z*(SE) = z*

You can plan for your study by having enough observations to guarantee a predetermined margin of error.

If you know what p is expected to be, you can use that value.

If not, you can use .5 as long as you have good evidence (CI) that p is between .3 and .7

n

pp )ˆ1(ˆ

Example

A college student organization wants to start a nightclub for students under the age of 21. To assess support for this proposal, they will select an SRS of students and ask each respondent if he or she would patronize this type of establishment. They expect that 70% of the student body would respond favorably. What sample size is required to obtain a 90% confidence interval with an approximate margin of error of 0.04? Suppose that 50% of the sample responds favorably?

Means or Proportions

Means It will be statedYou have quantitative data

ProportionsYou have some number out of a total

SRSYou have categorical data

Homework

Chapter 8

#28-30, 33, 36,40,44