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However, in the error bound formula, we use p ' q ' n as the standard deviation, instead of p q n

In the error bound formula, the sample proportions p ' and q ' are estimates of the unknown population proportions p and q . The estimated proportions p ' and q ' are used because p and q are not known. p ' and q ' are calculated from the data. p ' is the estimated proportion of successes. q ' is the estimated proportion of failures.

For the normal distribution of proportions, the z-score formula is as follows.

If P ' ~ N ( p , p q n ) then the z-score formula is z = p ' - p p q n

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 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 trueproportion of adults residents of this city who have cell phones.

Let X = the number of people in the sample who have cell phones. X is binomial. X ~ B ( 500 , 421 500 ) .

To calculate the confidence interval, you must find p ' , q ' , and EBP .

n = 500 x = the number of successes = 421

p ' = x n = 421 500 = 0.842

p ' = 0.842 is the sample proportion; this is the point estimate of the population proportion.

q ' = 1 - p ' = 1 - 0.842 = 0.158

Since CL = 0.95 , then α = 1 - CL = 1 - 0.95 = 0.05 α 2 = 0.025 .

z α 2 = z .025 = 1.96

Use the TI-83, 83+ or 84+ calculator command invnorm(.975,0,1) to find z .025 . Remember that the area to the right of z .025 is 0.025 and the area to the left of z .025 is 0.975. This can also be found using appropriate commands on other calculators, using a computer, or using a Standard Normal probability table.

EBP = z α 2 p ' q ' n = 1.96 [ ( .842 ) ( .158 ) 500 ] = 0.032

p ' - EBP = 0.842 - 0.032 = 0.81

p ' + EBP = 0.842 + 0.032 = 0.874

The confidence interval for the true binomial population proportion is ( p ' - EBP , p ' + EBP ) = ( 0.810 , 0.874 ) .

Interpretation

We estimate with 95% confidence that between 81% and 87.4% of all adult residents of this city have cell phones.

Explanation of 95% confidence level

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.

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 500students 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 confidenceinterval.

x = 300 and n = 500 . Using a TI-83+ or 84 calculator, the 90% confidence interval for the true percent of students that are registered voters is (0.564, 0.636).

p ' = x n = 300 500 = 0.600

q ' = 1 - p ' = 1 - 0.600 = 0.400

Since CL = 0.90 , then α = 1 - CL = 1 - 0.90 = 0.10 α 2 = 0.05 .

z α 2 = z .05 = 1.645

Use the TI-83, 83+ or 84+ calculator command invnorm(.95,0,1) to find z .05 . Remember that the area to the right of z .05 is 0.05 and the area to the left of z .05 is 0.95. This can also be found using appropriate commands on other calculators, using a computer, or using a Standard Normal probability table.

EBP = z α 2 p ' q ' n = 1.645 [ ( .60 ) ( .40 ) 500 ] = 0.036

p ' - EBP = 0.60 - 0.036 = 0.564

p ' + EBP = 0.60 + 0.036 = 0.636

    Interpretation:

  • We estimate with 90% confidence that the true percent of all students that are registered voters is between 56.4% and 63.6%.
  • Alternate Wording: We estimate with 90% confidence that between 56.4% and 63.6% of ALL students are registered voters.

Explanation of 90% confidence level

90% of all confidence intervals constructed in this way contain the true value for the population percent of students that are registered voters.

Calculating the sample size

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 proportion is EBP = z α 2 p ' q ' n . Solving for n gives you an equation for the sample size:

n z 2 p' q' EBP 2 , where z = z α 2

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 z α 2 = z .05 = 1.645 because the confidence level is 90%

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.

n z 2 p' q' EBP 2 gives n 1.645 2 (.5) (.5) .03 2 =751.7

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.

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Source:  OpenStax, Collaborative statistics: custom version modified by r. bloom. OpenStax CNX. Nov 15, 2010 Download for free at http://legacy.cnx.org/content/col10617/1.4
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