<< Chapter < Page Chapter >> Page >
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.

Confidence intervals ii

Confidence intervals for means

In the preceding considerations ( Confidence Intervals I ), the confidence interval for the mean μ of a normal distribution was found, assuming that the value of the standard deviation σ is known. However, in most applications, the value of the standard deviation σ is rather unknown, although in some cases one might have a very good idea about its value.

Suppose that the underlying distribution is normal and that σ 2 is unknown. It is shown that given random sample X 1 , X 2 , ... , X n from a normal distribution, the statistic T = X ¯ μ S / n has a t distribution with r = n 1 degrees of freedom, where S 2 is the usual unbiased estimator of σ 2 , (see, t distribution ).

Select t α / 2 ( n 1 ) so that P [ T t α / 2 ( n 1 ) ] = α / 2. Then

1 α = P [ t α / 2 ( n 1 ) X ¯ μ S / n t α / 2 ( n 1 ) ] = P [ t α / 2 ( n 1 ) S n X ¯ μ t α / 2 ( n 1 ) S n ] = P [ X ¯ t α / 2 ( n 1 ) S n μ X ¯ + t α / 2 ( n 1 ) S n ] = P [ X ¯ t α / 2 ( n 1 ) S n μ X ¯ + t α / 2 ( n 1 ) S n ] .

Thus the observations of a random sample provide a x ¯ and s 2 and x ¯ t α / 2 ( n 1 ) s n , x ¯ + t α / 2 ( n 1 ) s n is a 100 ( 1 α ) % interval for μ .

Let X equals the amount of butterfat in pound produced by a typical cow during a 305-day milk production period between her first and second claves. Assume the distribution of X is N ( μ , σ 2 ) . To estimate μ a farmer measures the butterfat production for n-20 cows yielding the following data:

481 537 513 583 453 510 570
500 487 555 618 327 350 643
499 421 505 637 599 392 -

For these data, x ¯ = 507.50 and s = 89.75 . Thus a point estimate of μ is x ¯ = 507.50 . Since t 0.05 ( 19 ) = 1.729 , a 90% confidence interval for μ is 507.50 ± 1.729 ( 89.75 20 ) , or equivalently, [472.80, 542.20].

Got questions? Get instant answers now!

Let T have a t distribution with n -1 degrees of freedom. Then, t α / 2 ( n 1 ) > z α / 2 . Consequently, the interval x ¯ ± z α / 2 σ / n is expected to be shorter than the interval x ¯ ± t α / 2 ( n 1 ) s / n . After all, there gives more information, namely the value of σ , in construction the first interval. However, the length of the second interval is very much dependent on the value of s . If the observed s is smaller than σ , a shorter confidence interval could result by the second scheme. But on the average, x ¯ ± z α / 2 σ / n is the shorter of the two confidence intervals.

If it is not possible to assume that the underlying distribution is normal but μ and σ are both unknown, approximate confidence intervals for μ can still be constructed using T = X ¯ μ S / n , which now only has an approximate t distribution.

Generally, this approximation is quite good for many normal distributions, in particular, if the underlying distribution is symmetric, unimodal, and of the continuous type. However, if the distribution is highly skewed , there is a great danger using this approximation. In such a situation, it would be safer to use certain nonparametric method for finding a confidence interval for the median of the distribution.

Confidence interval for variances

The confidence interval for the variance σ 2 is based on the sample variance S 2 = 1 n 1 i = 1 n ( X i X ¯ ) 2 .

In order to find a confidence interval for σ 2 , it is used that the distribution of ( n 1 ) S 2 / σ 2 is χ 2 ( n 1 ) . The constants a and b should selected from tabularized Chi Squared Distribution with n -1 degrees of freedom such that P ( a ( n 1 ) S 2 σ 2 b ) = 1 α .

That is select a and b so that the probabilities in two tails are equal: a = χ 1 α / 2 2 ( n 1 ) and b = χ α / 2 2 ( n 1 ) . Then, solving the inequalities, we have 1 α = P ( a ( n 1 ) S 2 1 σ 2 b ( n 1 ) S 2 ) = P ( ( n 1 ) S 2 b σ 2 ( n 1 ) S 2 a ) .

Thus the probability that the random interval  [(n-1)S 2 /b, (n-1)S 2 /a] contains the unknown σ 2 is 1- α . Once the values of X 1 , X 2 , ... , X n are observed to be x 1 , x 2 , ... , x n and s 2 computed, then the interval [(n-1)S 2 /b, (n-1)S 2 /a] is a 100 ( 1 α ) % confidence interval for σ 2 .

It follows that [ ( n 1 ) / b s , ( n 1 ) / a s ] is a 100 ( 1 α ) % confidence interval for σ , the standard deviation.

Assume that the time in days required for maturation of seeds of a species of a flowering plant found in Mexico is N ( μ , σ 2 ) . A random sample of n =13 seeds, both parents having narrow leaves, yielded x ¯ =18.97 days and 12 s 2 = i = 1 13 ( x x ¯ ) 2 = 128.41 .

A confidence interval for σ 2 is [ 128.41 21.03 , 128.41 5.226 ] = [ 6.11 , 24.57 ] , because 5.226 = χ 0.95 2 ( 12 ) and 21.03 = χ 0.055 2 ( 12 ) , what can be read from the tabularized Chi Squared Distribution. The corresponding 90% confidence interval for σ is [ 6.11 , 24.57 ] = [ 2.47 , 4.96 ] .

Got questions? Get instant answers now!

Although a and b are generally selected so that the probabilities in the two tails are equal, the resulting 100 ( 1 α ) % confidence interval is not the shortest that can be formed using the available data. The tables and appendixes gives solutions for a and b that yield confidence interval of minimum length for the standard deviation.

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
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play




Source:  OpenStax, Introduction to statistics. OpenStax CNX. Oct 09, 2007 Download for free at http://cnx.org/content/col10343/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Introduction to statistics' conversation and receive update notifications?

Ask