# 2.4 The t distribution

 Page 1 / 1
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.

## The t distribution

In probability and statistics, the t-distribution or Student's distribution arises in the problem of estimating the mean of a normally distributed population when the sample size is small, as well as when (as in nearly all practical statistical work) the population standard deviation is unknown and has to be estimated from the data.

## Textbook problems treating the standard deviation as if it were known are of two kinds:

• those in which the sample size is so large that one may treat a data-based estimate of the variance as if it were certain,
• those that illustrate mathematical reasoning, in which the problem of estimating the standard deviation is temporarily ignored because that is not the point that the author or instructor is then explaining.

## The t distribution

t Distribution
If Z is a random variable that is $N\left(0,1\right)$ , if U is a random variable that is ${\chi }^{2}\left(r\right)$ , and if Z and U are independent, then
$T=\frac{Z}{\sqrt{U/r}}=\frac{\overline{X}-\mu }{S/\sqrt{n}}$
has a t distribution with r degrees of freedom.

Where $\mu$ is the population mean, $\overline{x}$ is the sample mean and s is the estimator for population standard deviation (i.e., the sample variance) defined by

${s}^{2}=\frac{1}{N-1}{\sum _{i=1}^{N}\left({x}_{i}-\overline{x}\right)}^{2}.$

If $\sigma =s$ , $t=z$ , the distribution becomes the normal distribution. As N increases, Student’s t distribution approaches the normal distribution . It can be derived by transforming student’s z -distribution using $z\equiv \frac{\overline{x}-\mu }{s}$ and then defining $t=z\sqrt{n-1}.$

The resulting probability and cumulative distribution functions are:

$f\left(t\right)=\frac{\Gamma \left[\left(r+1\right)/2\right]}{\sqrt{\pi r}\Gamma \left(r/2\right){\left(1+{t}^{2}/r\right)}^{\left(r+1\right)/2}},$
$F\left(t\right)=\frac{1}{2}+\frac{1}{2}\left[I\left(1;\frac{1}{2}r,\frac{1}{2}\right)-I\left(\frac{r}{r+{t}^{2}},\frac{1}{2}r,\frac{1}{2}\right)\right]\mathrm{sgn}\left(t\right)=\frac{1}{2}-\frac{itB\left(-\frac{{t}^{2}}{r};\frac{1}{2},\frac{1}{2}\left(1-r\right)\right)\Gamma \left(\frac{1}{2}\left(r+1\right)\right)}{2\sqrt{\pi }|t|\Gamma \left(\frac{1}{2}r\right)}$

where,

• $r=n-1$ is the number of degrees of freedom,
• $-\infty
• $\Gamma \left(z\right)$ is the gamma function,
• $B\left(a,b\right)$ is the bets function,
• $I\left(z;a,b\right)$ is the regularized beta function defined by $I\left(z;a,b\right)=\frac{B\left(z;a,b\right)}{B\left(a,b\right)}.$

The effect of degree of freedom on the t distribution is illustrated in the four t distributions on the Figure 1 .

In general, it is difficult to evaluate the distribution function of T . Some values are usually given in the tables. Also observe that the graph of the p.d.f. of T is symmetrical with respect to the vertical axis t =0 and is very similar to the graph of the p.d.f. of the standard normal distribution $N\left(0,1\right)$ . However the tails of the t distribution are heavier that those of a normal one; that is, there is more extreme probability in the t distribution than in the standardized normal one. Because of the symmetry of the t distribution about t =0, the mean (if it exists) must be equal to zero. That is, it can be shown that $E\left(T\right)=0$ when $r\ge 2$ . When r =1 the t distribution is the Cauchy distribution , and thus both the variance and mean do not exist.

how can chip be made from sand
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
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
has a lot of application modern world
Kamaluddeen
yes
narayan
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!

#### Get Jobilize Job Search Mobile App in your pocket Now! By Robert Murphy By Steve Gibbs By OpenStax By Steve Gibbs By Vanessa Soledad By Briana Hamilton By OpenStax By Richley Crapo By Darlene Paliswat By JavaChamp Team