7.3 Confidence interval for a population proportion  (Page 2/7)

 Page 2 / 7

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

The confidence interval can only be used if the number of successes $np\text{'}$ and the number of failures $nq\text{'}$ are both larger than 5.

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

If $P\text{'}$ ~ $N\left(p,\sqrt{\frac{p\cdot q}{n}}\right)$ then the z-score formula is $z=\frac{p\text{'}-p}{\sqrt{\frac{p\cdot q}{n}}}$

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

Solution

• You can use technology to directly calculate the confidence interval.
• The first solution is step-by-step (Solution A).
• The second solution uses a function of the TI-83, 83+ or 84 calculators (Solution B).

Using a function of the ti-83, 83+ or 84 calculators:

Press STAT and arrow over to TESTS .
Arrow down to A:1-PropZint . Press ENTER .
Arrow down to $x$ and enter 421.
Arrow down to $n$ and enter 500.
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 estimate 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.

• You can use technology to directly calculate the confidence interval.
• The first solution is step-by-step (Solution A).
• The second solution uses a function of the TI-83, 83+ or 84 calculators (Solution B).

Solution a

$x=300$ and $n=500$ .

$p\text{'}=\frac{x}{n}=\frac{300}{500}=0.600$

$q\text{'}=1-p\text{'}=1-0.600=0.400$

Since $\text{CL}=0.90$ , then $\alpha =1-\text{CL}=1-0.90=0.10\phantom{\rule{20pt}{0ex}}\frac{\alpha }{2}=0.05$ .

${z}_{\frac{\alpha }{2}}={z}_{.05}=1.645\phantom{\rule{20pt}{0ex}}$

Use the TI-83, 83+ or 84+ calculator command invNorm(0.95,0,1) 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.

$\text{EBP}={z}_{\frac{\alpha }{2}}\cdot \sqrt{\frac{p\text{'}\cdot q\text{'}}{n}}=1.645\cdot \sqrt{\frac{\left(0.60\right)\cdot \left(0.40\right)}{500}}=0.036$

$p\text{'}-\text{EBP}=0.60-0.036=0.564$

$p\text{'}+\text{EBP}=0.60+0.036=0.636$

The confidence interval for the true binomial population proportion is $\left(p\text{'}-\text{EBP},p\text{'}+\text{EBP}\right)=$ $\left(0.564,0.636\right)$ .

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.

Solution b

Using a function of the TI-83, 83+ or 84 calculators:

Press STAT and arrow over to TESTS .
Arrow down to A:1-PropZint . Press ENTER .
Arrow down to $x$ and enter 300.
Arrow down to $n$ and enter 500.
Arrow down to C-Level and enter .90.
Arrow down to Calculate and press ENTER .
The confidence interval is (0.564, 0.636).

Calculating the sample size n

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

• $\mathrm{EBP}={z}_{\frac{\alpha }{2}}\cdot \sqrt{\frac{\mathrm{p\text{'}q\text{'}}}{n}}$
• Solving for $n$ gives you an equation for the sample size.
• $n=\frac{{{z}_{\frac{\alpha }{2}}}^{2}\cdot \mathrm{p\text{'}q\text{'}}}{{\mathrm{EBP}}^{2}}$

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.

Solution

From the problem, we know that EBP=0.03 (3%=0.03) and

${z}_{\frac{\alpha }{2}}={z}_{.05}=1.645$ 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.

$n=\frac{z^{2}\mathrm{p\text{'}}\mathrm{q\text{'}}}{\mathrm{EBP}^{2}}$ gives $n=\frac{\mathrm{1.645}^{2}\mathrm{\left(.5\right)}\mathrm{\left(.5\right)}}{\mathrm{.03}^{2}}$ =751.7

Round the answer to the next higher value. The sample size should be 752 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.

what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!          By