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The “plus four” method for calculating confidence intervals is an attempt to balance the error introduced by using estimates of the population proportion when calculating the standard deviation of the sampling distribution. Simply imagine four additional trials in the study; two are successes and two are failures. Calculate p = x + 2 n + 4 , and proceed to find the confidence interval. When sample sizes are small, this method has been demonstrated to provide more accurate confidence intervals than the standard formula used for larger samples.

Formula review

p′ = x / n where x represents the number of successes and n represents the sample size. The variable p ′ is the sample proportion and serves as the point estimate for the true population proportion.

q ′ = 1 – p

p ~ N ( p , p q n ) The variable p′ has a binomial distribution that can be approximated with the normal distribution shown here.

EBP = the error bound for a proportion = z α 2 p q n

Confidence interval for a proportion:

( lower bound, upper bound) = ( p E B P , p + E B P ) = ( p z p q n ,   p + z p q n )

n =   z α 2 2 p q E B P 2 provides the number of participants needed to estimate the population proportion with confidence 1 - α and margin of error EBP .

Use the normal distribution for a single population proportion p   = x n

E B P = ( z α 2 ) p q n   p + q = 1

The confidence interval has the format ( p′ EBP , p′ + EBP ).

x ¯ is a point estimate for μ

p′ is a point estimate for ρ

s is a point estimate for σ

Use the following information to answer the next two exercises: Marketing companies are interested in knowing the population percent of women who make the majority of household purchasing decisions.

When designing a study to determine this population proportion, what is the minimum number you would need to survey to be 90% confident that the population proportion is estimated to within 0.05?

If it were later determined that it was important to be more than 90% confident and a new survey were commissioned, how would it affect the minimum number you need to survey? Why?

It would decrease, because the z-score would decrease, which reducing the numerator and lowering the number.


Use the following information to answer the next five exercises: Suppose the marketing company did do a survey. They randomly surveyed 200 households and found that in 120 of them, the woman made the majority of the purchasing decisions. We are interested in the population proportion of households where women make the majority of the purchasing decisions.

Identify the following:

  1. x = ______
  2. n = ______
  3. p′ = ______

Define the random variables X and P′ in words.

X is the number of “successes” where the woman makes the majority of the purchasing decisions for the household. P ′ is the percentage of households sampled where the woman makes the majority of the purchasing decisions for the household.

Which distribution should you use for this problem?

Construct a 95% confidence interval for the population proportion of households where the women make the majority of the purchasing decisions. State the confidence interval, sketch the graph, and calculate the error bound.

CI: (0.5321, 0.6679)

This is a normal distribution curve. The peak of the curve coincides with the point 0.6 on the horizontal axis.  A central region is shaded between points 0.5321 and 0.6679.

EBM : 0.0679

Questions & Answers

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s. Reply
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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.
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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.
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what is the Synthesis, properties,and applications of carbon nano chemistry
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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is Bucky paper clear?
CYNTHIA
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s. Reply
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.
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Do you know which machine is used to that process?
s.
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for screen printed electrodes ?
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s. Reply
of graphene you mean?
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or in general
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in general
s.
Graphene has a hexagonal structure
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Cied
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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anybody can imagine what will be happen after 100 years from now in nano tech world
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Statistics i - math1020 - red river college - version 2015 revision a - draft 2015-10-24. OpenStax CNX. Oct 24, 2015 Download for free at http://legacy.cnx.org/content/col11891/1.8
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