Chapter seventeen: The chi-square
statistic: tests for goodness of fit
Introduction to the chi Square:
and independence
The test for goodness of it
Parametric, and non-parametrics II. The null hypothesis can state
the proportions for one
statistical test
population are not different
Parametric tests:Test evaluating from The proportions than are
hypotheses about population
known to exist for another
parameters and making
population.
assumptions about parameters.
Simple data:
Also, a test requiring numerical
Just select a sample of n
scores.
individuals and count how many
are in each category.
The resulting values are called
observed frequencies
Sample distributions
Non-barometric test:
When experimental situations
do not conform to the
requirements of the parametric
test, a non-barometric test
The chi squared test for
may be used.
goodness of fit Continued.
in most situations, the
The overall goal is to compare
barometric test is preferred
the data, which is the observed
because it is more likely to
frequencies with the null
detect a real difference or a
hypothesis. The first step is to
real relationship. However, if it
construct a hypothetical sample
is easier to transform the
that represents how the
score into categories, a nonsample distribution looks if it
parametric test may be
were in a perfect agreement
preferred. This can include if
stated in the null hypothesis.
it’s easier to obtain category
Expected frequency: For each
category is the frequency value
measurements, or if the
that is predicted from the
original scores may violate the
proportions in the null
assumptions. Additionally, in
the original score has unusually hypothesis and the sample size.
high variance a non-parametric
test is preferred, and
occasionally an experiment
Step one: find the difference
produces an undetermined or
between the data and the
infinite score when for
hypothesis for each category.
example, participant fails to
Step two: the difference,
solve a problem You can say
ensuring that all values are
that the participant is in the
positive.
highest category and then
Step three: divide the squared
classify the other scores
difference by the hypothesis.
according to their numerical
Step four: sum the values from
values.
all the categories
The chi-square test for
The Chi squared, distribution
goodness-of-fit
and degrees of freedom
Uses sample data to test
To decide whether a value is
hypotheses about the shape or
larger small, we must refer to a
proportions of a population
chi square distribution.
distribution. It will determine
This distribution is a set of chi
how well the obtain sample
square
values for all the
proportions fit the population
possible
random samples, when
proportions, specified by the
HO is true.
null hypothesis.
This distribution is a
I. The null hypothesis often
theoretical distribution with
states there is no preference
well defined characteristics.
among the different
The formula involves adding
categories. In this case it
squared values, so you can never
states that the population is
obtain a negative value so all
divided equally among
values are zero or larger.
When HO is true, you expect the
FO values to be close to the FE
values. Thus we expect the Chi
square values to be small. These
two factors suggest that the
distribution will be positively
skewed.
Locating the critical region
To determine if a Chi square value
is significantly large. You must
consult the chi square
distribution table.
The Chi square test for
independence
Uses the frequency data from a
sample to evaluate the
relationship between two
variables in the population.
Each individual in the sample is
classified on both of the two
variables, creating a two
dimensional, frequency,
distribution matrix.
The frequency distribution
for the sample is used to test
hypothesis about the
corresponding frequency
distribution in the population.
The null hypothesis states that
the two variables being
measured are independent, and
can be expressed in two forms,
each viewing the data and the
test from slightly different
perspectives.
I. The data are viewed as a single
sample with each individual
measured on two variables,
stating that there is no
relationship between the two
variables.
II. The data are viewed as two or
more separate samples,
representing two or more
populations or treatment
conditions. The goal of the Chi
squared test is to determine
whether there are significant
differences between the
populations, stating that there
is no difference between the
two populations.