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(What does "Endorsed by" mean?)Summary: Confidence Interval for a Population Proportion is part of the collection col10555 written by Barbara Illowsky and Susan Dean with contributions from Roberta Bloom.
During an election year, we see articles in the newspaper that state confidence intervals
in terms of proportions or percentages. For example, a poll for a particular
candidate running for president might show that the candidate has 40% of the vote
within 3 percentage points. Often, election polls are calculated with 95% confidence.
So, the pollsters would be 95% confident that the true proportion of voters who
favored the candidate would be between 0.37 and 0.43 :
Investors in the stock market are interested in the true proportion of stocks that go up and down each week. Businesses that sell personal computers are interested in the proportion of households in the United States that own personal computers. Confidence intervals can be calculated for the true proportion of stocks that go up or down each week and for the true proportion of households in the United States that own personal computers.
The procedure to find the confidence interval, the sample size, the error bound, and the confidence level for a proportion is similar to that for the population mean. The formulas are different.
How do you know you are dealing with a proportion problem? First, the
underlying distribution is binomial. (There is no mention of a mean or average.) If
(Sometimes the random variable is denoted as
When
If we divide the random variable by
Using algebra to simplify :
The confidence interval has the form
The error bound for a proportion is
This formula is similar to the error bound formula for a
mean, except that the "appropriate standard deviation" is different. For a mean, when the population standard deviation is known, the appropriate standard deviation that we use is
However, in the error bound formula, we use
However, in the error bound formula, the standard deviation is
In the error bound formula, the sample proportions
If
Suppose that a market research firm is hired to estimate the percent of adults living in a large city who have cell phones. 500 randomly selected adult residents in this city are surveyed to determine whether they have cell phones. Of the 500 people surveyed, 421 responded yes - they own cell phones. Using a 95% confidence level, compute a confidence interval estimate for the true proportion of adults residents of this city who have cell phones.
Let
To calculate the confidence interval, you must find
Since
Then
Use the TI-83, 83+ or 84+ calculator command invnorm(.975,0,1) to find
The confidence interval for the true binomial population proportion is
We estimate with 95% confidence that between 81% and 87.4% of all adult residents of this city have cell phones.
95% of the confidence intervals constructed in this
way would contain the true value for the population proportion of all adult residents of this city who have cell phones.
Press STAT and arrow over to TESTS.
Arrow down to A:PropZint. Press ENTER.
Arrow down to
Arrow down to
Arrow down to C-Level and enter .95.
Arrow down to Calculate and press ENTER.
The confidence interval is (0.81003, 0.87397).
For a class project, a political science student at a large university
wants to determine the percent of students that are registered voters. He surveys 500
students and finds that 300 are registered voters. Compute a 90% confidence interval
for the true percent of students that are registered voters and interpret the confidence
interval.
Since
Use the TI-83, 83+ or 84+ calculator command invnorm(.95,0,1) to find
The confidence interval for the true binomial population proportion is
90% of all confidence intervals constructed in this way contain the true value for the population percent of students that are registered voters.
Using a function of the TI-83, 83+ or 84 calculators:
Press STAT and arrow over to TESTS.
Arrow down to A:PropZint. Press ENTER.
Arrow down to
Arrow down to
Arrow down to C-Level and enter .90.
Arrow down to Calculate and press ENTER.
The confidence interval is (0.564, 0.636).
If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.
The error bound formula for a population proportion is
Suppose a mobile phone company wants to determine the current percentage of customers aged 50+ that use text messaging on their cell phone. How many customers aged 50+ should the company survey in order to be 90% confident that the estimated (sample) proportion is within 3 percentage points of the true population proportion of customers aged 50+ that use text messaging on their cell phone.
From the problem, we know that
EBP=0.03 (3%=0.03) and
However, in order to find n , we need to know the estimated (sample) proportion p'. Remember that q'=1-p'. But, we do not know p' yet. Since we multiply p' and q' together, we make them both equal to 0.5 because p'q'= (.5)(.5)=.25 results in the largest possible product. (Try other products: (.6)(.4)=.24; (.3)(.7)=.21; (.2)(.8)=.16 and so on). The largest possible product gives us the largest n. This gives us a large enough sample so that we can be 90% confident that we are within 3 percentage points of the true population proportion. To calculate the sample size n, use the formula and make the substitutions.
Round the answer to the next higher value. The sample size should be 758 cell phone customers aged 50+ in order to be 90% confident that the estimated (sample) proportion is within 3 percentage points of the true population proportion of all customers aged 50+ that use text messaging on their cell phone.
**With contributions from Roberta Bloom.
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Last edited by Susan Dean on Aug 19, 2011 10:28 am GMT-5.
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