chi square

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CHI SQUARE TEST
Tests for Goodness of Fit
and Independence

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Preview
Color is known to affect human moods and
emotion. Sitting in a pale-blue room is more
calming than sitting in a bright-red room
Based on the known influence of color, Hill
and Barton (2005) hypothesized that the
color of uniform may influence the outcome
of physical sports contest
The study does not produce a numerical
score for each participant. Each participant
is simply classified into two categories
(winning or losing)

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Preview
The data consist of frequencies or proportions
describing how many individuals are in each
category
This study want to use a hypothesis test to
evaluate data. The null hypothesis would state
that color has no effect on the outcome of the
contest
Statistical technique have been developed
specifically to analyze and interpret data
consisting of frequencies or proportions
CHI SQUARE

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PARAMETRIC AND NONPARAMETRI
STATISTICAL TESTS
The tests that concern parameter and
require assumptions about parameter are
called parametric tests
Another general characteristic of parametric
tests is that they require a numerical score
for each individual in the sample. In terms
of measurement scales, parametric tests
require data from an interval or a ratio scale

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PARAMETRIC AND NONPARAMETRI
STATISTICAL TESTS
Often, researcher are confronted with
experimental situation that do not conform to
the requirements of parametric tests. In this
situations, it may not be appropriate to use a
parametric test because may lead to an
erroneous interpretation of the data
Fortunately, there are several hypothesis
testing techniques that provide alternatives to
parametric test that called nonparametric tests

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NONPARAMETRIC TEST
Nonparametric tests sometimes are called
distribution free tests
One of the most obvious differences
between parametric and nonparametric
tests is the type of data they use
All the parametric tests required numerical
scores. For nonparametric, the subjects are
usually just classified into categories

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NONPARAMETRIC TEST
Notice that these classification involve
measurement on nominal or ordinal scales,
and they do not produce numerical values
that can be used to calculate mean and
variance
Nonparametric tests generally are not as
sensitive as parametric test; nonparametric
tests are more likely to fail in detecting a
real difference between two treatments

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THE CHI SQUARE TEST FOR GOODNESS O
… uses sample data to test hypotheses about
the shape or proportions of a population
distribution. The test determines how well the
obtained sample proportions fit the
population proportions specified by the null
hypothesis

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THE NULL HYPOTHESIS FOR THE GOODNESS O
For the chi-square test of goodness of fit, the
null hypothesis specifies the proportion (or
percentage) of the population in each category
Generally H 0 will fall into one of the following
categories:
○No preference
H0 states that the population is divided equally
among the categories
○No difference from a Known population
H0 states that the proportion for one population are
not different from the proportion that are known to
exist for another population

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THE DATA FOR THE GOODNESS OF FIT TE
Select a sample of n individuals and count how
many are in each category
The resulting values are called observed
frequency (f
o
)
A sample of n = 40 participants was given a
personality questionnaire and classified into
one of three personality categories: A, B, or C
Category A Category B Category C
15 19 6

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EXPECTED FREQUENCIES
The general goal of the chi-square test for goodness
of fit is to compare the data (the observed
frequencies) with the null hypothesis
The problem is to determine how well the data fit the
distribution specified in H 0 – hence name goodness of
fit
Suppose, for example, the null hypothesis states that
the population is distributed into three categories
with the following proportion
Category A Category B Category C
25% 50% 25%

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EXPECTED FREQUENCIES
To find the exact frequency expected for each
category, multiply the same size (n) by the
proportion (or percentage) from the null
hypothesis
25% of 40 = 10 individual in category A
50% of 40 = 20 individual in category B
25% of 40 = 10 individual in category C

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THE CHI-SQUARE STATISTIC
The general purpose of any hypothesis test
is to determine whether the sample data
support or refute a hypothesis about
population
In the chi-square test for goodness of fit, the
sample expressed as a set of observe
frequencies (fovalues) and the null
hypothesis is used to generate a set of
expected frequencies (f
evalues)

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THE CHI-SQUARE STATISTIC
The chi-square statistic simply measures ho
well the data (fo
) fit the hypothesis (fe
)
The symbol for the chi-square statistic is χ2
The formula for the chi-square statistic is
χ2
= ∑
(fo – fe)2
fe

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A researcher has developed three different
design for a computer keyboard. A sample of n
= 60 participants is obtained, and each
individual tests all three keyboard and identifies
his or her favorite.
The frequency distribution of preference is:
Design A = 23, Design B = 12, Design C = 25.
Use a chi-square test for goodness of fit with α
= .05 to determine whether there are significant
preferences among three design
LEARNING CHECK

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Dari https://twitter.com/#!/palangmerah
diketahui bahwa persentase golongan darah
di Indonesia adalah:
A : 25,48%,
B : 26,68%,
O : 40,77 %,
AB : 6,6 %
Golongan darah di kelas kita?
Apakah berbeda dengan data PMI?
LEARNING CHECK

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THE CHI-SQUARE TEST FOR INDEPENDE
The chi-square may also be used to test
whether there is a relationship between two
variables
For example, a group of students could be
classified in term of personality (introvert,
extrovert) and in terms of color preferences
(red, white, green, or blue).
RED WHITE GREEN BLUE ∑
INTRO 10 3 15 22 50
EXTRO 90 17 25 18 150
100 20 40 40 200

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OBSERVED AND EXPECTED FREQUEN
fo
RED WHITE GREEN BLUE ∑
INTRO 10 3 15 22 50
EXTRO 90 17 25 18 150
∑ 100 20 40 40 200
fe
RED WHITE GREEN BLUE ∑
INTRO 50
EXTRO 150
∑ 100 20 40 40 200

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OBSERVED AND EXPECTED FREQUEN
fo
RED WHITE GREEN BLUE ∑
INTRO 10 3 15 22 50
EXTRO 90 17 25 18 150
∑ 100 20 40 40 200
fe
RED WHITE GREEN BLUE ∑
INTRO 25 5 10 10 50
EXTRO 75 15 30 30 150
∑ 100 20 40 40 200

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OBSERVED AND EXPECTED FREQUEN
fo R W G B ∑
INTRO 10 3 15 22 50
EXTRO 90 17 25 18 150
∑ 100 20 40 40 200
(fo– fe)2
R W G B
INTRO
EXTRO
fe R W G B ∑
INTRO 25 5 10 10 50
EXTRO 75 15 30 30 150
∑ 100 20 40 40 200

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OBSERVED AND EXPECTED FREQUEN
fo R W G B ∑
INTRO 10 3 15 22 50
EXTRO 90 17 25 18 150
∑ 100 20 40 40 200
(fo– fe)2
R W G B
INTRO (-15) 2
(-2)2
(5)2
(12)2
EXTRO (15)2
(-2)2
(-5)2
(-12)2
fe R W G B ∑
INTRO 25 5 10 10 50
EXTRO 75 15 30 30 150
∑ 100 20 40 40 200

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OBSERVED AND EXPECTED FREQUEN
(fo– fe)2
/fe R W G B
INTRO
EXTRO
fe R W G B
INTRO 25 5 10 10
EXTRO 75 15 30 30
(fo– fe)2
R W G B
INTRO 225 4 25 144
EXTRO 225 4 25 144

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OBSERVED AND EXPECTED FREQUEN
(fo– fe)2
/fe R W G B
INTRO 9 0,8 2,5 14,4
EXTRO 3 0,267 0,833 4,8
fe R W G B
INTRO 25 5 10 10
EXTRO 75 15 30 30
(fo– fe)2
R W G B
INTRO 225 4 25 144
EXTRO 225 4 25 144

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THE CHI-SQUARE STATISTIC
χ2
= ∑
(fo – fe)2
fe
χ2
= 35,6
df = (C-1) (R-1) = (3) (1) = 3
χ2
critical at α = .05 is 7,81