10-3Estimating a Population Proportion

Recall from Chapter 9...

Remember n is the sample size and p is the population parameter.

So...

Note: You ALWAYS have to check the conditions when creating a CI, even if the problem doesn’t ask you to do so!!

Standard Error

Since we don’t know p, we will use phat to estimate p.

The standard deviation will be called the Standard Error (SE) of phat.

Think of SE as an estimate of the standard deviation of the sampling distribution.

(1 )p pSE

n

Using :

Find z* for a 95% CI Find z* for a 99% CIFind z* for a 70% CIFind z* for a 85% CI

1 C /100invNorm

2

A confidence Interval for a ProportionOne-proportion Z Interval

In May 2002, the Gallup Poll asked 537 randomly sampled adults the question “Generally speaking, do you believe the death penalty is applied fairly or unfairly in this country today?” Of these, 53% answered “Fairly” and 7% said they didn't know.

What can we conclude from this survey?

Construct a 95% confidence interval for the proportion p of U.S. Adults who think the death

penalty is applied fairly.

A confidence Interval for a ProportionOne-proportion Z Interval

In May 2002, the Gallup Poll asked 537 randomly sampled adults the question “Generally speaking, do you believe the death penalty is applied fairly or unfairly in this country today?” Of these, 53% answered “Fairly” and 7% said they didn't know.

What can we conclude from this survey?

Steps from p.631 Inference Toolbox1) Parameter2) Conditions3) Calculations

4) Interpretation “3 C’s”: conclusion, connection, & context

Binge DrinkingIn a 2001 survey of 10,904 US college undergraduate students, 2486 students were classified as binge drinkers

Calculate a 99% CI. We are 99% confident that the proportion of college undergraduates who engaged in frequent binge drinking lies between 0.218 and 0.238.

How Proportion CI’s Behave

The margin of error gets smaller when:

• z* gets smaller (Confidence level decreases)

• phat gets further from 0.5

• n gets larger

(1 )p pSE

n

Caution: Other Types of Error

• Remember that the margin of error in this confidence interval includes only random sampling error! That is error due to chance variation in randomized data production.

• Other sources of error (referred to as non-sampling error) such as response bias, nonresponse, undercoverage, designs other than SRS that we treat like an SRS, etc. are not included in our margin of error.

More Cautions

• Data must be an SRS from the population. Different methods are needed for different sampling designs. We will only study CI’s from an SRS.

• There is no correct method for inference from data haphazardly collected with bias of unknown size. GIGO

• Outliers can distort results.

• The shape of the population matters. Check your conditions!

• The size of the sample determines the margin of error, not the size of the population (as long as the N ≥ 10n).

Using the Calculator

Calculating a confidence interval for a population proportion using the calculator

Stat tests A:1-PropZInt

x = number of successesn = sample sizeC-Level: enter the confidence level as an integer such as 90, 95, or 99 Press enter to Calculate

Note: Write m as a decimal.

Determining Sample Size

A company has received complaints about it s customer service. They intend to hire a

consultant to carry out a survey of customers. Before contacting the consultant, the company

president wants some idea of the sample size that she will be required to pay for. The president

wants to estimate the proportion p of customers who are satisfied. She decides she wants the estimate to be within 3% at a 95% confidence

interval. Approximate an appropriate sample size.

Determining Sample Size

A company has received complaints about it s customer service. They intend to hire a

consultant to carry out a survey of customers. Before contacting the consultant, the company

president wants some idea of the sample size that she will be required to pay for. The president

wants to estimate the proportion p of customers who are satisfied. She decides she wants the estimate to be within 3% at a 95% confidence

interval. Approximate an appropriate sample size.

Remember to always round n up!