# 13.2 Simple random samples and statistics

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The (simple) random sample, is basic to much of classical statistics. Once formulated, we may apply probability theory to exhibit several basic ideas of statistical analysis. A population may be most any collection of individuals or entities. Associated with each member is a quantity or a feature that can be assigned a number. The population distribution is the distribution of that quantity among the members of the population.To obtain information about the population distribution, we select “at random” a subset of the population and observe how the quantity varies over the sample. Hopefully, the distribution in the sample will give a useful approximation to the population distribution. We obtain values of such quantities as the mean and variance in the sample (which are random quantities) and use these as estimators for corresponding population parameters (which are fixed). Probability analysis provides estimates of the variation of the sample parameters about the corresponding population parameters.

## Simple random samples and statistics

We formulate the notion of a (simple) random sample , which is basic to much of classical statistics. Once formulated, we may apply probability theory to exhibitseveral basic ideas of statistical analysis.

We begin with the notion of a population distribution . A population may be most any collection of individuals or entities. Associated with each member is aquantity or a feature that can be assigned a number. The quantity varies throughout the population. The population distribution is the distribution of that quantityamong the members of the population.

If each member could be observed, the population distribution could be determined completely. However, that is not always feasible. In order to obtain informationabout the population distribution, we select “at random” a subset of the population and observe how the quantity varies over the sample. Hopefully, thesample distribution will give a useful approximation to the population distribution.

The sampling process

We take a sample of size n , which means we select n members of the population and observe the quantity associated with each. The selection is done in such a manner thaton any trial each member is equally likely to be selected. Also, the sampling is done in such a way that the result of any one selection does not affect, and is not affected by,the others. It appears that we are describing a composite trial. We model the sampling process as follows:

• Let X i , $1\le i\le n$ be the random variable for the i th component trial. Then the class $\left\{{X}_{i}:1\le i\le n\right\}$ is iid, with each member having the population distribution.

This provides a model for sampling either from a very large population (often referred to as an infinite population) or sampling with replacement from a small population.

The goal is to determine as much as possible about the character of the population. Two important parameters are the mean and the variance. We want the population mean and thepopulation variance. If the sample is representative of the population, then the sample mean and the sample variance should approximate the population quantities.

#### Questions & Answers

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
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
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive