Statistics Dictionary
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ChiSquare Goodness of Fit Test
A chisquare goodness of fit test attempts
to answer the following question: Are sample data
consistent with a hypothesized distribution?
The test is appropriate when the following conditions are met:

The sampling method is
simple random sampling
.
 The population is at least 10 times as large as the sample.

The variable under study is
categorical
.

The expected value for each
level
of the variable is at least 5.
Here is how to conduct the test.

Define hypotheses. For a chisquare goodness of fit test,
the hypotheses take the following form.
H_{0}: The data are consistent with a specified distribution.
H_{a}: The data are not consistent with a
specified distribution.

Typically, the null hypothesis specifies the proportion of
observations at each level of the categorical variable. The
alternative hypothesis is that at least one of
the specified proportions is not true.

Specify significance level. Often, researchers choose
significance levels
equal to
0.01, 0.05, or 0.10; but any value between 0 and
1 can be used.

Find degrees of freedom. The
degrees of freedom
(DF) is equal to the
number of levels (k) of the categorical variable minus one:
DF = k  1 .

Compute expected frequency counts. The expected frequency counts
at each level of the categorical variable are equal to
the sample size times the hypothesized proportion
from the null hypothesis
E_{i} = np_{i}
where
E_{i} is the expected frequency count for the
ith level of the categorical variable,
n is the total sample size, and
p_{i} is the hypothesized proportion of observations
in level i.

Find test statistic. The test statistic is a chisquare random variable
(Χ^{2}) defined by
the following equation.
Χ^{2} =
Σ [ (O_{i}  E_{i})^{2} / E_{i} ]
where
O_{i} is the observed frequency count for the
ith level of the categorical variable, and
E_{i} is the expected frequency count for the
ith level of the categorical variable.

Find Pvalue.
The Pvalue is the probability of observing a
sample statistic as extreme as the test statistic. Since the
test statistic is a chisquare, use the
ChiSquare Distribution Calculator
to assess the probability associated with the test statistic. Use
the degrees of freedom computed above.
If the sample findings are unlikely, given
the null hypothesis, the researcher rejects the null hypothesis.
Typically, this involves comparing the Pvalue to the
significance level
,
and rejecting the null hypothesis when the Pvalue is less than
the significance level.