# 11.3 Goodness-of-fit test

This module describes how the chi-square distribution is used to conduct goodness-of-fit test.

In this type of hypothesis test, you determine whether the data "fit" a particular distribution or not. For example, you may suspect your unknown data fit a binomialdistribution. You use a chi-square test (meaning the distribution for the hypothesis test is chi-square) to determine if there is a fit or not. The null and the alternate hypotheses for this test may be written in sentences or may be stated as equations orinequalities.

The test statistic for a goodness-of-fit test is:

$\underset{k}{\Sigma }\frac{\left(O-E{\right)}^{2}}{E}$

where:

• $O$ = observed values (data)
• $E$ = expected values (from theory)
• $k$ = the number of different data cells or categories

The observed values are the data values and the expected values are the values you would expect to get if the null hypothesis were true. There are $n$ terms of the form $\frac{\left(O-E{\right)}^{2}}{E}$ .

The degrees of freedom are $\text{df = (number of categories - 1)}$ .

The goodness-of-fit test is almost always right tailed. If the observed values and the corresponding expected values are not close to each other, then the test statisticcan get very large and will be way out in the right tail of the chi-square curve.

The expected value for each cell needs to be at least 5 in order to use this test.

Absenteeism of college students from math classes is a major concern to math instructors because missing class appears to increase the drop rate. Suppose that a study was done to determine if the actual student absenteeism follows faculty perception. The faculty expected that a group of 100 students would miss class according to the following chart.

 Number absences per term Expected number of students 0 - 2 50 3 - 5 30 6 - 8 12 9 - 11 6 12+ 2

A random survey across all mathematics courses was then done to determine the actual number (observed) of absences in a course. The next chart displays the result of that survey.

 Number absences per term Actual number of students 0 - 2 35 3 - 5 40 6 - 8 20 9 - 11 1 12+ 4

Determine the null and alternate hypotheses needed to conduct a goodness-of-fit test.

${H}_{o}$ : Student absenteeism fits faculty perception.

The alternate hypothesis is the opposite of the null hypothesis.

${H}_{a}$ : Student absenteeism does not fit faculty perception.

Can you use the information as it appears in the charts to conduct the goodness-of-fit test?

No. Notice that the expected number of absences for the "12+" entry is less than 5 (it is 2). Combine that group with the "9 - 11" group to create new tables where the number of students for each entry are at least 5. The new tables are below.

 Number absences per term Expected number of students 0 - 2 50 3 - 5 30 6 - 8 12 9+ 8
 Number absences per term Actual number of students 0 - 2 35 3 - 5 40 6 - 8 20 9+ 5

What are the degrees of freedom ( $\mathrm{df}$ )?

There are 4 "cells" or categories in each of the new tables.

$\mathrm{df = number of cells - 1 = 4 - 1 = 3}$

Employers particularly want to know which days of the week employees are absent in a five day work week. Most employers wouldlike to believe that employees are absent equally during the week. Suppose a random sample of 60 managers were asked on which day of the week did they have the highest number of employee absences. The results were distributed as follows:

Day of the week employees were most absent
Monday Tuesday Wednesday Thursday Friday
Number of Absences 15 12 9 9 15

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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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1 It is estimated that 30% of all drivers have some kind of medical aid in South Africa. What is the probability that in a sample of 10 drivers: 3.1.1 Exactly 4 will have a medical aid. (8) 3.1.2 At least 2 will have a medical aid. (8) 3.1.3 More than 9 will have a medical aid.