<< Chapter < Page Chapter >> Page >

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

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
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.
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!

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
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Collaborative statistics (custom lecture version modified by t. short)' conversation and receive update notifications?