Section 8.3 ~ Estimating Population Proportions

Introduction to Probability and StatisticsMs. Young

Objective

Sec. 8.3

After this section you will learn how to estimate population proportions and compute the associated margins of error and confidence intervals.

The Basics of Estimating a Population Proportion

The process for estimating a population proportion, p, with a 95% confidence level using a sample proportion, , is the same as the process of estimating a population mean using a sample mean (section 8.2) The only difference is the way that the margin of error is defined:

The confidence interval is written as:

Sec. 8.3

p̂

ˆ ˆ(1 )2E

n

ˆ ˆE E

or

ˆ E

Example 1

Sec. 8.3

The Nielsen ratings for television use a random sample of households. A Nielsen survey results in an estimate that a women’s World Cup soccer game had 72.3% of the entire viewing audience. Assuming that the sample consists of n = 5,000 randomly selected households, find the margin of error and the 95% confidence interval for this estimate.

The 95% confidence interval is:

0.723 – 0.013 < p < 0.723 + 0.013

or

0.710 < p < 0.736

With 95% confidence, we can conclude that between 71% and 73.6% of the entire viewing audience watched the women’s World Cup soccer game.

ˆ ˆ(1 )2E

n

0.723(1 0.723) 2

5000E

0.013E

Choosing Sample Size Choosing a sample size appropriate to satisfy a desired margin of

error is found by manipulating this APPROXIMATE formula for margin of error:

Note: any value equal to or larger than the value found using the formula would be sufficient

Sec. 8.3

1 E

n 1E n

1 n

E

21

nE

2

1 n

E

Used to approximate appropriate sample size

2

1 n

E

Example 2

Sec. 8.3

You plan a survey to estimate the proportion of students on your campus who carry a cell phone regularly. How many students should be in the sample if you want (with 95% confidence) a margin of error of no more than 4 percentage points?

You should survey at least 625 students.

2

1n

E

2

1

0.04n

1 625

.0016n