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This module provides the assumptions to be considered in order to calculate a Test of Two Variances and how to execute the Test of Two Variances. An example is provided to help clarify the concept.

Another of the uses of the F distribution is testing two variances. It is often desirable to compare two variances rather than two averages. For instance, collegeadministrators would like two college professors grading exams to have the same variation in their grading. In order for a lid to fit a container, the variation in the lidand the container should be the same. A supermarket might be interested in the variability of check-out times for two checkers.

In order to perform a F test of two variances, it is important that the following are true:

  1. The populations from which the two samples are drawn are normally distributed.
  2. The two populations are independent of each other.

Suppose we sample randomly from two independent normal populations. Let σ 1 2 and σ 2 2 be the population variances and s 1 2 and s 2 2 be the sample variances. Let the sample sizes be n 1 and n 2 . Since we are interested in comparing the two sample variances, we use the F ratio

F = [ ( s 1 ) 2 ( σ 1 ) 2 ] [ ( s 2 ) 2 ( σ 2 ) 2 ]

F has the distribution F ~ F ( n 1 - 1 , n 2 - 1 )

where n 1 - 1 are the degrees of freedom for the numerator and n 2 - 1 are the degrees of freedom for the denominator.

If the null hypothesis is σ 1 2 = σ 2 2 , then the F-Ratio becomes F = [ ( s 1 ) 2 ( σ 1 ) 2 ] [ ( s 2 ) 2 ( σ 2 ) 2 ] = ( s 1 ) 2 ( s 2 ) 2 .

The F ratio could also be ( s 2 ) 2 ( s 1 ) 2 . It depends on H a and on which sample variance is larger.

If the two populations have equal variances, then s 1 2 and s 2 2 are close in value and F = ( s 1 ) 2 ( s 2 ) 2 is close to 1 . But if the two population variances are very different, s 1 2 and s 2 2 tend to be very different, too.Choosing s 1 2 as the larger sample variance causes the ratio ( s 1 ) 2 ( s 2 ) 2 to be greater than 1 . If s 1 2 and s 2 2 are far apart, then F = ( s 1 ) 2 ( s 2 ) 2 is a large number.

Therefore, if F is close to 1 , the evidence favors the null hypothesis (the two population variances are equal). But if F is much larger than 1 , then the evidence is against the null hypothesis.

A test of two variances may be left, right, or two-tailed.

Two college instructors are interested in whether or not there is any variation in the way they grade math exams. They each grade the same set of 30exams. The first instructor's grades have a variance of 52.3. The second instructor's grades have a variance of 89.9.

Test the claim that the first instructor's variance is smaller. (In most colleges, it is desirable for the variances of exam grades to be nearlythe same among instructors.) The level of significance is 10%.

Let 1 and 2 be the subscripts that indicate the first and second instructor, respectively.

n 1 = n 2 = 30 .

H o : σ 1 2 = σ 2 2 and H a : σ 1 2 σ 2 2

Calculate the test statistic: By the null hypothesis ( σ 1 2 = σ 2 2 ) , the F statistic is

F = [ ( s 1 ) 2 ( σ 1 ) 2 ] [ ( s 2 ) 2 ( σ 2 ) 2 ] = ( s 1 ) 2 ( s 2 ) 2 = 52.3 89.9 = 0.5818

Distribution for the test: F 29 , 29 where n 1 - 1 = 29 and n 2 - 1 = 29 .

Graph: This test is left tailed.

Draw the graph labeling and shading appropriately.

Probability statement: p-value = P ( F 0.5818 ) = 0.0753

Compare α and the p-value: α = 0.10 α > p-value .

Make a decision: Since α > p-value , reject H o .

Conclusion: With a 10% level of significance, from the data, there is sufficient evidence to conclude that the variance in grades for the first instructor is smaller.

TI-83+ and TI-84: Press STAT and arrow over to TESTS . Arrow down to D:2-SampFTest . Press ENTER . Arrow to Stats and press ENTER . For Sx1 , n1 , Sx2 , and n2 , enter ( 52.3 ) , 30 , ( 89.9 ) , and 30 . Press ENTER after each. Arrow to σ1: and σ2 . Press ENTER . Arrow down to Calculate and press ENTER . F = 0.5818 and p-value = 0.0753 . Do the procedure again and try Draw instead of Calculate .

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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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
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Professor
I think
Professor
what is the stm
Brian Reply
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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?
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What is meant by 'nano scale'?
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What is STMs full form?
LITNING
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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what is differents between GO and RGO?
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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 .
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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 ?
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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Source:  OpenStax, Collaborative statistics: custom version modified by r. bloom. OpenStax CNX. Nov 15, 2010 Download for free at http://legacy.cnx.org/content/col10617/1.4
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