# 13.4 Test of two variances

 Page 1 / 1
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 ${\sigma }_{1}^{2}$ and ${\sigma }_{2}^{2}$ 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

$F=\frac{\left[\frac{\left({s}_{1}{\right)}^{2}}{{\left({\sigma }_{1}\right)}^{2}}\right]}{\left[\frac{\left({s}_{2}{\right)}^{2}}{{\left({\sigma }_{2}\right)}^{2}}\right]}$

$F$ has the distribution $F$ ~ $F\left({n}_{1}-1,{n}_{2}-1\right)$

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 ${\sigma }_{1}^{2}={\sigma }_{2}^{2}$ , then the F-Ratio becomes $F=\frac{\left[\frac{\left({s}_{1}{\right)}^{2}}{{\left({\sigma }_{1}\right)}^{2}}\right]}{\left[\frac{\left({s}_{2}{\right)}^{2}}{{\left({\sigma }_{2}\right)}^{2}}\right]}=\frac{\left({s}_{1}{\right)}^{2}}{{\left({s}_{2}\right)}^{2}}$ .

The $F$ ratio could also be $\frac{\left({s}_{2}{\right)}^{2}}{{\left({s}_{1}\right)}^{2}}$ . 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 $F=$ $\frac{\left({s}_{1}{\right)}^{2}}{{\left({s}_{2}\right)}^{2}}$ 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 $\frac{\left({s}_{1}{\right)}^{2}}{{\left({s}_{2}\right)}^{2}}$ to be greater than $1$ . If ${s}_{1}^{2}$ and ${s}_{2}^{2}$ are far apart, then $F=$ $\frac{\left({s}_{1}{\right)}^{2}}{{\left({s}_{2}\right)}^{2}}$ 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.

${n}_{1}={n}_{2}=30$ .

${H}_{o}$ : ${\sigma }_{1}^{2}={\sigma }_{2}^{2}$ and ${H}_{a}$ : ${\sigma }_{1}^{2}$ $()$ ${\sigma }_{2}^{2}$

Calculate the test statistic: By the null hypothesis $\left({\sigma }_{1}^{2}={\sigma }_{2}^{2}\right)$ , the F statistic is

$F=\frac{\left[\frac{\left({s}_{1}{\right)}^{2}}{{\left({\sigma }_{1}\right)}^{2}}\right]}{\left[\frac{\left({s}_{2}{\right)}^{2}}{{\left({\sigma }_{2}\right)}^{2}}\right]}=\frac{\left({s}_{1}{\right)}^{2}}{{\left({s}_{2}\right)}^{2}}=\frac{52.3}{89.9}=0.5818$

Distribution for the test: ${F}_{29,29}\phantom{\rule{20pt}{0ex}}$ where ${n}_{1}-1=29$ and ${n}_{2}-1=29$ .

Graph: $\phantom{\rule{20pt}{0ex}}$ This test is left tailed.

Draw the graph labeling and shading appropriately.

Probability statement: $\text{p-value}=P$ ( $F$ $()$ $0.5818$ ) $=0.0753$

Compare $\alpha$ and the p-value: $\alpha =0.10$ $\phantom{\rule{20pt}{0ex}}\alpha >\text{p-value}$ .

Make a decision: Since $\alpha >\text{p-value}$ , 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  $\sqrt{\left(52.3\right)}$  , 30 ,  $\sqrt{\left(89.9\right)}$  , and 30 . Press ENTER after each. Arrow to σ1: and  $()$ σ2 . Press ENTER . Arrow down to Calculate and press ENTER . $F=0.5818$ and $\text{p-value}=0.0753$ . Do the procedure again and try Draw instead of Calculate .

#### Questions & Answers

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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Got questions? Join the online conversation and get instant answers!     By By Anonymous User By Lakeima Roberts   