# 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

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Stoney Reply
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Adin Reply
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Kyle
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biomolecules are e building blocks of every organics and inorganic materials.
Joe
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research.net
kanaga
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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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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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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
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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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
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s. Reply
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.
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s.
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for screen printed electrodes ?
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What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
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in general
s.
Graphene has a hexagonal structure
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Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
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