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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.

Questions & Answers

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write examples of Nano molecule?
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The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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what king of growth are you checking .?
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yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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why?
Adin
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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Introduction about quantum dots in nanotechnology
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absolutely yes
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Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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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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