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General form of a confidence interval

( lower value , upper value ) = ( point estimate - margin of error , point estimate + margin of error )

To find the error bound when you know the confidence interval

margin of error = upper value - point estimate OR margin of error = upper value - lower value 2

Single population mean, known standard deviation, normal distribution

Use the Normal Distribution for Means ME = z α 2 σ n

The confidence interval has the format ( x - ME , x + ME ) .

Single population mean, unknown standard deviation, student's-t distribution

Use the Student's-t Distribution with degrees of freedom df = n - 1 . ME = t α 2 s n

Single population proportion, normal distribution

Use the Normal Distribution for a single population proportion p ̂ = x n

ME = z α 2 p ̂ q ̂ n p ̂ + q ̂ = 1

The confidence interval has the format ( p ̂ - ME , p ̂ + ME ) .

Point estimates

x is a point estimate for μ

p ̂ is a point estimate for ρ

s is a point estimate for σ

Critical values

z* is a critical value, based on the Normal Curve for an exact confidence interval such as for a 95% confidence interval. z* = 1.96

t* is a critical value, based on the Student's-t for an exact confidence interval.


Binomial distribution

A discrete random variable (RV) which arises from Bernoulli trials. There are a fixed number, n , of independent trials. “Independent” means that the result of any trial (for example, trial 1) does not affect the results of the following trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV X size 12{X} {} is defined as the number of successes in n trials. The notation is: X ~ B ( n , p ) . The mean is μ np and the standard deviation is σ = npq . The probability of exactly x successes in n trials is P ( X = x ) = n x p x q n x .

Confidence interval (ci)

An interval estimate for an unknown population parameter. This depends on:
(1). The desired confidence level.
(2). Information that is known about the distribution (for example, known standard deviation).
(3). The sample and its size.

Confidence level

The percent expression for the probability that the confidence interval contains the true population parameter. For example, if the CL=90%, then in 90 out of 100 samples the interval estimate will enclose the true population parameter.

Degrees of freedom (df)

The number of objects in a sample that are free to vary.

Margin of error for a population mean (ebm)

The margin of error. Depends on the confidence level, sample size, and known or estimated population standard deviation.

Margin of error for a population proportion (ebp)

The margin of error. Depends on the confidence level, sample size, and the estimated (from the sample) proportion of successes.

Inferential statistics

Also called statistical inference or inductive statistics. This facet of statistics deals with estimating a population parameter based on a sample statistic. For example, if 4 out of the 100 calculators sampled are defective we might infer that 4 percent of the production is defective

Normal distribution

A continuous random variable (RV) with pdf f(x) = 1 σ e ( x μ ) 2 / 2 size 12{ ital "pdf"= { {1} over {σ sqrt {2π} } } e rSup { size 8{ - \( x - μ \) rSup { size 6{2} } /2σ rSup { size 6{2} } } } } {} , where μ is the mean of the distribution and σ is the standard deviation. Notation: X ~ N μ σ . If μ = 0 and σ = 1 , the RV is called the standard normal distribution .


A numerical characteristic of the population.

Point estimate

A single number computed from a sample and used to estimate a population parameter.

Student-t distribution

Investigated and reported by William S. Gossett in 1908 and published under the pseudonym Student. The major characteristics of the random variable (RV) are:
(1). It is continuous and assumes any real values.
(2). The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution.
(3). It approaches the standard normal distribution as n gets larger.
(4). There is a "family" of t distributions: every representative of the family is completely defined by the number of degrees of freedom which is one less than the number of data

Standard deviation

A number that is equal to the square root of the variance and measures how far data values are from their mean. Notation: s for sample standard deviation and σ for population standard deviation.

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
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