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Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution). Using a subscript that matches the random variable, suppose:

  • μ X = the mean of X
  • σ X = the standard deviation of X
If you draw random samples of size n , then as n increases, the random variable X which consists of sample means, tends to be normally distributed and

X ~ N ( μ X , σ X n )

The Central Limit Theorem for Sample Means says that if you keep drawing larger and larger samples (like rolling 1, 2, 5, and, finally, 10 dice) and calculating their means the sample means form their own normal distribution (the sampling distribution). The normal distribution has the same mean as the original distribution and a variance that equals the original variance divided by n , the sample size. n is the number of values that are averaged together not the number of times the experiment is done.

To put it more formally, if you draw random samples of size n ,the distribution of the random variable X , which consists of sample means, is called the sampling distribution of the mean . The sampling distribution of the mean approaches a normal distribution as n , the sample size, increases.

The random variable X has a different z-score associated with it than the random variable X . x is the value of X in one sample.

z = x - μ X ( σ X n )

μ X is both the average of X and of X .

σ X = σ X n = standard deviation of X and is called the standard error of the mean.

An unknown distribution has a mean of 90 and a standard deviation of 15. Samples of size n = 25 are drawn randomly from the population.

Find the probability that the sample mean is between 85 and 92.

Let X = one value from the original unknown population. The probability question asks you to find a probability for the sample mean .

Let X = the mean of a sample of size 25. Since μ X = 90 , σ X = 15 , and n = 25 ;

then X ~ N ( 90 , 15 25 )

Find P ( 85 x 92 ) Draw a graph.

P ( 85 x 92 ) = 0.6997

The probability that the sample mean is between 85 and 92 is 0.6997.

Normal distribution curve from -∞ to ∞ and an x-axis with the values of 85, 90, and 92. The x-axis is equal to the mean of a sample size of 25. A vertical upward line extends from points 85 and 92 to the curve. The probability area is between 85 and 92.

TI-83 or 84: normalcdf (lower value, upper value, mean, standard error of the mean)

The parameter list is abbreviated (lower value, upper value, μ , σ n )

normalcdf (85,92,90, 15 25 ) = 0.6997

Find the value that is 2 standard deviations above the expected value (it is 90) of the sample mean.

To find the value that is 2 standard deviations above the expected value 90, use the formula

value = μ X + (#ofSTDEVs) ( σ X n )

value = 90 + 2 15 25 = 96

So, the value that is 2 standard deviations above the expected value is 96.

The length of time, in hours, it takes an "over 40" group of people to play one soccer match is normally distributed with a mean of 2 hours and a standard deviation of 0.5 hours . A sample of size n = 50 is drawn randomly from the population.

Find the probability that the sample mean is between 1.8 hours and 2.3 hours.

Let X = the time, in hours, it takes to play one soccer match.

The probability question asks you to find a probability for the sample mean time, in hours , it takes to play one soccer match.

Let X = the mean time, in hours, it takes to play one soccer match.

If μ X = _________, σ X = __________, and n = ___________, then X ~ N (______, ______) by the Central Limit Theorem for Means.

μ X = 2 , σ X = 0.5 , n = 50 , and X ~ N ( 2 , 0.5 50 )

Find P ( 1.8 x 2.3 ) . Draw a graph.

P ( 1.8 x 2.3 ) = 0.9977

normalcdf (1.8,2.3,2, .5 50 ) = 0.9977

The probability that the mean time is between 1.8 hours and 2.3 hours is ______.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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Source:  OpenStax, Collaborative statistics (custom lecture version modified by t. short). OpenStax CNX. Jul 15, 2013 Download for free at http://cnx.org/content/col11543/1.1
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