1. 8.3 Estimating Population Proportions
LEARNING GOAL
Learn to estimate population proportions and compute the associated margins of error and confidence intervals.
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The Basics of Estimating a Population Proportion
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95% Confidence Interval for a Population Proportion
For a population proportion, the margin of error for the 95% confidence interval is
where is the sample proportion.
The 95% confidence interval ranges
from – margin of error to margin of error
We can write this confidence interval more formally as
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EXAMPLE 2 TV Nielsen Ratings
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. ˆ p
Solution: The sample proportion, = 72.3% = 0.723, is the best estimate of the population proportion.
The margin of error is
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EXAMPLE 2 TV Nielsen Ratings
Solution: (cont.)
The 95% confidence interval is
0.723 – < p < , or
With 95% confidence, we conclude that between 71.0% and 73.6% of the entire viewing audience watched the women’s World Cup soccer game.
0.710 < p < 0.736
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Choosing Sample Size
Choosing the Correct Sample Size
In order to estimate a population proportion with a 95% degree of confidence and a specified margin of error of E, the size of the sample should be at least
n = 1
E 2
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EXAMPLE 4 Minimum Sample Size for Survey
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?
Solution: Note that 4 percentage points means a margin of error of From the given formula, the minimum sample size is
You should survey at least 625 students. 1
E 2 1
0.04 2
n = = = 625
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The End
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