6.3 Confidence Intervals for Population Proportions

Statistics

Mrs. Spitz

Spring 2009

Objectives/Assignment

How to find a sample proportion How to construct a confidence interval for a

population proportion How to determine a minimum sample size

when estimating a population proportion.

Assignment: pp. 280-282 #1-27 all

Schedule for coming weeks:

Today – Notes 6.3. Homework due BOC on Friday.

Friday, 1/16/09 – Notes 6.4. Assignment due Tuesday on our return.

Monday – 1/19/09 – No school Tuesday – 1/20/09 – Chapter Review Thursday-Chapter Review 6 DUE – Test –

Chapter 6 Friday – 1/23/09 – 7.1 Hypothesis Testing

Sample Proportions

Recall from section 4.2 that the probability of success in a single trial of a binomial experiment is p. This probability is a population proportion. In this section, you will learn how to estimate a population proportion, p using a confidence interval. As with confidence intervals for µ, you will start with a point estimate (6.1)

Definition:

The point estimate for p, the population proportion of successes, is given by the proportion of successes in a sample and is denoted by:

n

xp ˆ

where x is the number of successes in the sample and n is the number in the sample. The point estimate for the number of failures is .The symbols and are read as “p hat” and “q hat”

pq ˆ1ˆ q̂ p̂

Ex. 1: Finding a point estimate for p

In a survey of 883 American adults, 380 said that their favorite sport is football. Find a point estimate for the population proportion of adults who say their favorite sport is football.

SOLUTION: Using n =883 and x = 380

%4343.0883

380ˆ n

xp

Insight

In the first two sections, estimates were made for the quantitative data. In this section, sample proportions are used to make estimates for qualitative data.

Confidence Intervals for a Population P

Constructing a confidence interval for a population proportion p is similar to constructing a confidence interval for a population mean. You start with a point estimate and calculate a maximum error of estimate.

EppEp ˆˆ

Definition:

A c-confidence interval for the population proportion p is

where

n

qpzE c

ˆˆ

The probability that the confidence interval contains p is c.

EppEp ˆˆ

Notes

In section 5.5, you learned that a binomial can be approximated by the normal distribution if np 5 and nq 5. When and , the sampling distribution for is approximately normal with a mean of p = p and a standard error of

n

pqp

5ˆ pn5ˆ qn p̂

Guidelines: Constructing a Confidence Interval for a Population Proportion

In words1. ID the sample stats, n and x2. Find the point estimate3. Verify the sampling

distribution of p(hat) can be approximated by the normal distribution.

4. Find the critical zc that corresponds to the given level of confidence, c.

5. Find the maximum error of estimate, E.

6. Find the left and the right endpoints and form the confidence interval.

n

xp ˆ

Is and is ?

Use a standard normal table.

n

qpzE c

ˆˆ

Left endpoint:

Right endpoint:

Interval:

5ˆ pn 5ˆ qn

EppEp ˆˆ

Ep ˆEp ˆ

Ex. 2: Constructing a Confidence interval for p Construct a 95% confidence interval for the

proportion of American adults who say that their favorite sport is football.

SOLUTION: Form example 1, , So,

. Using n = 883, you can verify that the sampling distribution of can be approximated by the normal distribution.

and

43.0ˆ p57.043.01ˆ q

p̂

538043.0883ˆ pn

550357.0883ˆ qn

Ex. 2: Constructing a Confidence interval for pUsing zc = 1.96, the maximum error of estimate is:

033.0883

)57.0)(43.0(96.1

ˆˆ

n

qpzE c

The 95% confidence interval is as follows:

397.0033.043.0ˆ EpLeft Endpoint Right Endpoint

463.0033.043.0ˆ Ep

463.0397.0 p

So, with 95% confidence, you can say that the proportion of adults who say that footbal is their favorite sport is between 39.7% and 46.3%.

Opinion Polls

The confidence level of 95% used in Example 2 is typical of opinion polls. The result; however, is usually not stated as a confidence interval. Instead the result of Example 2 would usually be stated as 43% with a margin of error of 3.3%.”

Ex. 3: Constructing a Confidence Interval for p

The graph shown below is from a survey of 935 adults. Construct a 99% confidence interval for the proportion of adults who think that airplanes are the safest mode of transportation.

Solution:

From the graph So, Using these values and the values n = 935 and zc = 2.575, the maximum error of estimate is:

.45.0ˆ p 55.045.01ˆ q

042.0935

)55.0)(45.0(575.2

ˆˆ

n

qpzE c

The 99% confidence inteval is as follows:

408.0042.045.0ˆ EpLeft Endpoint Right Endpoint

492.0042.045.0ˆ Ep

492.0408.0 p

So with 99% confidence, you can say that the proportion of adults who think that airplanes are the safest mode of transportation is between 40.8% and 49.2%

Increasing Sample Size to Increase Precision

One way to increase the precision of the confidence interval without decreasing the level of confidence is to increase the sample size.

Insight – why 0.5?

The reason for using 0.5 as values for p hat and q hat when no preliminary estimate is available is that these values yield a maximum value for the product

In other words, if you don’t estimate thevalues of p hat and q hat, you must pay the penalty of using a larger sample.

)ˆ1(ˆˆˆ ppqp

Ex. 4: Determining a Minimum Sample Size

You are running a political campaign and wish to estimate with 95% confidence, the proportion of registered voters, who will vote for your candidate. What is the minimum sample size needed if you are to accurately within 3% of the population proportion?

SOLUTION

Because you do not have a preliminary estimate for p, use and . Using zc = 1.96, and E = 0.03, you can solve for n.

5.0ˆ p 5.0ˆ q

2

ˆˆE

zqpn c 11.1067

03.0

96.1)5.0)(5.0(

2

Because n is a decimal, round up to the nearest whole number. So, at least 1068 registered voters should be included in the sample.

Assignment due Friday BOC.

Assignment: pp. 280-282 #1-22 all