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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

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
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
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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