Inference for a Population ProportionSection 12.1

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Remember Conditions for Inference

Data are an SRS from the population of interest.

Observations are independent (pop. ≥ 10*n)

Sampling Distribution is approx. normal Today, we’re dealing with proportions, so np ≥ 10 and n(1- p) ≥ 10.

Standard Error Replace standard deviation by the standard

error of (or standard deviation of )

To get a confidence interval of the form

Estimate ± z* SE

Inference for a Population ProportionDraw an SRS of size n from a large population with unknown proportion p of successes. An approximate level C confidence interval for p is

where z* is the upper (1 – C)/2 standard normal critical value.

Remember: State Plan Do Conclude

Statistics Problems Demand Consistency!!!

Example 1A Gallup Poll found that 28% of a SRS of 682 American adults expect to inherit money. Construct a 90% Confidence interval for the true proportion.

State: know what parameters we’re estimating & at what confidence level

We want to estimate p = the true proportion of US adults who expect to inherit $ with 90% confidence.

Example 1Plan: choose method & check conditions

Method: Proportions

Conditions:Random:

Independent:

Normal:

Assume Gallup used correct sampling procedures

n = 682, the population of adults is much larger than 6820 (pop. ≥ 10*n), so assume independence.

sampling distribution of is approx. normal

Example 1Do: if conditions are met, perform calculations

.

Example 1Conclude: interpret the interval in the context of the problem

We are 90% confident that the true percentage is between 25.17% and 30.83%.

YOUR TURN!!!The New York Times and CBS News conducted a nationwide poll of 1048 randomly selected 13- to 17-year-olds. Of these teenagers, 692 had a television in their room. We will act as if the sample were an SRS.

Construct a 95% confidence interval for the proportion of all people in this age group who have a TV in their room.

!!!!We are trying to estimate the population proportion of teenagers who have a TV in their room at a 95% confidence level.

Method: proportions, Conditions: SRS: Yes!

Independent: Population of teenagers ≥ 10*1048 Yes! Normal: (1048)(.66) ≈ 692 ≥ 10 and (1048)(.34) ≈ 356 ≥ 10

Yes!

We are 95% confident that the true population proportion of teenagers with a TV in their room falls between .63 and .69.

Choosing the sample size

Since the margin of error contains the sample proportion, we need to guess this value when choosing n.

We will call this guess p*.

Choosing the sample sizeTwo ways to get p*:

1. Use p* based on a past experience with similar studies. Cover several calculations to cover the range of -values you might find.

Better to use when you have done a similar study.

2. Use p* = 0.5 as the guess. The margin of error m is largest when . Use when you suspect to be between 0.3 and 0.7

Choosing the sample sizeSo…

Where p* is a guessed value for the sample proportion.

Example 12.9, p. 696 Gloria Chavez and Ronald Flynn are the

candidates for mayor in a large city. You are planning a sample survey to determine what percent of the voters plan to vote for Chavez. This is a population proportion p. You will contact an SRS of registered voters in the city. You want to estimate p with 95% confidence and a margin of error no greater than 3%, or 0.03. How large a sample do you need?

Example 12.9, p. 696Gloria Chavez and Ronald Flynn are the candidates for mayor in a large city. You are planning a sample survey to determine what percent of the voters plan to vote for Chavez. This is a population proportion p. You will contact an SRS of registered voters in the city. You want to estimate p with 95% confidence and a margin of error no greater than 3%, or 0.03. How large a sample do you need?

Should we use p* = 0.5? YES!

Gloria Chavez and Ronald Flynn are the candidates for mayor in a large city. You are planning a sample survey to determine what percent of the voters plan to vote for Chavez. This is a population proportion p. You will contact an SRS of registered voters in the city. You want to estimate p with 95% confidence and a margin of error no greater than 3%, or 0.03. How large a sample do you need?

So we want:

32.66 ≤ 1067.1≤ n

So we need n = 1068 to satisfy this inequality.

Homework: p. 694: 12.8, 12.9 P. 696: 12. 10, 12.11

Due: Tuesday