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This course is a short series of lectures on Introductory Statistics. Topics covered are listed in the Table of Contents. The notes were prepared by EwaPaszek and Marek Kimmel. The development of this course has been supported by NSF 0203396 grant.

Confidence intervals i

Given a random sample X 1 , X 2 ,..., X n from a normal distribution N ( μ , σ 2 ) , consider the closeness of X ¯ , the unbiased estimator of μ , to the unknown μ . To do this, the error structure (distribution) of X ¯ , namely that X ¯ is N ( μ , σ 2 / n ) , is used in order to construct what is called a confidence interval for the unknown parameter μ , when the variance σ 2 is known.

For the probability 1 α , it is possible to find a number z α / 2 , such that P ( z α / 2 X ¯ μ σ / n z α / 2 ) = 1 α .

For example , if 1 α = 0.95 , then z α / 2 = z 0.025 = 1.96 and if 1 α = 0.90 , then z α / 2 = z 0.05 = 1.645.

Recalling that σ > 0 , the following inequalities are equivalent : z α / 2 X ¯ μ σ / n z α / 2 and z α / 2 ( σ n ) X ¯ μ z α / 2 ( σ n ) ,

X ¯ z α / 2 ( σ n ) μ X ¯ + z α / 2 ( σ n ) , X ¯ + z α / 2 ( σ n ) μ X ¯ z α / 2 ( σ n ) .

Thus, since the probability of the first of these is 1- 1 α , the probability of the last must also be 1 α , because the latter is true if and only if the former is true. That is, P [ X ¯ z α / 2 ( σ n ) μ X ¯ + z α / 2 ( σ n ) ] = 1 α .

So the probability that the random interval [ X ¯ z α / 2 ( σ n ) , X ¯ + z α / 2 ( σ n ) ] includes the unknown mean μ is 1 α .

The number 100 ( 1 α ) % , or equivalently, 1 α , is called the confidence coefficient .

For illustration , x ¯ ± 1.96 ( σ / n ) is a 95% confidence interval for μ .

It can be seen that the confidence interval for μ is centered at the point estimate x ¯ and is completed by subtracting and adding the quantity z α / 2 ( σ / n ) .

as n increases, z α / 2 ( σ / n ) decreases, resulting n a shorter confidence interval with the same confidence coefficient 1 α

A shorter confidence interval indicates that there is more reliance in x ¯ as an estimate of μ . For a fixed sample size n , the length of the confidence interval can also be shortened by decreasing the confidence coefficient 1 α . But if this is done, shorter confidence is achieved by losing some confidence.

Let x ¯ be the observed sample mean of 16 items of a random sample from the normal distribution N ( μ , σ 2 ) . A 90% confidence interval for the unknown mean μ is [ x ¯ 1.645 23.04 16 , x ¯ + 1.645 23.04 16 ] . For a particular sample this interval either does or does not contain the mean μ . However, if many such intervals were calculated, it should be true that about 90% of them contain the mean μ .

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If one cannot assume that the distribution from which the sample arose is normal, one can still obtain an approximate confidence interval for μ . By the Central Limit Theorem the ratio ( X ¯ μ ) / ( σ / n ) has, provided that n is large enough, the approximate normal distribution N ( 0 , 1 ) when the underlying distribution is not normal. In this case P ( z α / 2 X ¯ μ σ / n z α / 2 ) 1 α , and [ x ¯ z α / 2 ( σ n ) , x ¯ + z α / 2 ( σ n ) ] is an approximate 100 ( 1 α ) % confidence interval for μ . The closeness of the approximate probability 1 α to the exact probability depends on both the underlying distribution and the sample size. When the underlying distribution is unimodal (has only one mode) and continuous, the approximation is usually quite good for even small n , such as n = 5 . As the underlying distribution becomes less normal ( i.e. , badly skewed or discrete), a larger sample size might be required to keep reasonably accurate approximation. But, in all cases, an n of at least 30 is usually quite adequate.

Confidence Intervals II

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
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 .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Introduction to statistics. OpenStax CNX. Oct 09, 2007 Download for free at http://cnx.org/content/col10343/1.3
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