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Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution) and 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 ΣX which consists of sums tends to be normally distributed and

Σ X ~ N ( n μ X , n σ X )

The Central Limit Theorem for Sums says that if you keep drawing larger and larger samples and taking their sums, the sums form their own normal distribution (the sampling distribution) which approaches a normal distribution as the sample size increases. The normal distribution has a mean equal to the original mean multiplied by the sample size and a standard deviationequal to the original standard deviation multiplied by the square root of the sample size.

The random variable Σ X has the following z-score associated with it:

  • Σx is one sum.
  • z = Σ x - n μ X n σ X
  • n μ X = the mean of ΣX
  • n σ X = standard deviation of ΣX

An unknown distribution has a mean of 90 and a standard deviation of 15. A sample of size 80 is drawn randomly from the population.

  • Find the probability that the sum of the 80 values (or the total of the 80 values) is more than 7500.
  • Find the sum that is 1.5 standard deviations above the mean of the sums.

Let X = one value from the original unknown population. The probability question asks you to find a probability for the sum (or total of) 80 values.

ΣX = the sum or total of 80 values. Since μ X = 90 , σ X = 15 , and n = 80 , then

Σ X ~ N ( 80 90 , 80 15 )

  • mean of the sums = n μ X = ( 80 ) ( 90 ) = 7200
  • standard deviation of the sums = n σ X = 80 15
  • sum of 80 values = Σx = 7500

  • Find P ( Σx 7500 )

P ( Σx 7500 ) = 0.0127

Normal distribution curve of sum X with the values of 7200 and 7500 on the x-axis. A vertical upward line extends from point 7500 on the x-axis up to the curve. The probability area occurs from point 7500 and to the end of the curve.

normalcdf (lower value, upper value, mean of sums, stdev of sums)

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

normalcdf (7500,1E99, 80 90 , 80 15 ) = 0.0127

Reminder: 1E99 = 10 99 . Press the EE key for E.

  • Find Σx where z = 1.5:


Σx = n μ X + z n σ X = (80)(90) + (1.5)( 80 ) (15)= 7401.2

Questions & Answers

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
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
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
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 (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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