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Chi-Square Tests
Chapter 13
The chi-square test for Goodness of Fit allows us to
determine whether a specified population distribution
seems valid.
The Chi-Square (
) test is an inferential test that
shows whether or not a frequency distribution fits an
expected or claimed distribution.
1. The chi-square distribution
is NOT symmetric.
2. The shape depends on the
degrees of freedom.
3. As the number of df
increases, the chi-square
distribution becomes more
symmetric. Otherwise, each
curve is skewed right.
4. All values are non-negative.
5. Chi-Square has df =
(number of categories) - 1
1st: State the hypothesis
Ho: Frequency fits a specified distribution (actual equals
hypothesized)
Ha: Frequency does not fit a specified distribution. (Actual is
different from hypothesized).
The observed frequency (O), of a category is the frequency
(count or value) of the category that is observed in the sample
data.
The expected frequency (E) of a category is the calculated
frequency obtained assuming that the null hypothesis is true.
(E=np)
n=sample size
p=probability
To use the chi-square goodness of fit test, the following
conditions must be met:
1. All observed data are obtained using a random sample.
2. All expected frequencies are greater than or equal to 1.
3. No more than 20% of the expected frequencies are less than
5.
O is the observed: Enter into L1
E is the expected: Enter into L2
L3=(L1-L2)^2/L2
For critical values, use Table C (Chi-Square
Distribution)
Calculator Commands:
Catalog, Sum (L3)---This is your chi-square value.
Distribution,
cdf(Ans, E99,df)---This is your p-value.
Chi-Squared Test of Independence
 A chi-squared two-way table test is a test that determines whether
two variables are:
Ho: Independent/ have no association.
Ha: Dependent/ have an association.
Conditions: Same as χ2 GOF test.
Data is randomly selected. All expected cell counts are at least
1 and no more than 20% of the expected cell counts are less
than 5.
df = (r-1)(c-1)
r= # of rows, c= # of columns
Do not include the “total” row/column.
Expected Cells
(𝑠𝑢𝑚 𝑜𝑓 𝑟𝑜𝑤)(𝑠𝑢𝑚 𝑜𝑓 𝑐𝑜𝑙𝑢𝑚𝑛)
 E=
𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒
Chi-Squared Test of Homogeneity
 This tests the claim that several proportions are equal when
samples are taken from different populations.
Ho: All proportions are equal.
Ha: At least one of the proportions is different from the others.
df = (r-1)(c-1)
Conditions: Same as other Chi-squared tests.