Chapter 26
Chi-Square Testing
Chi-Square Testing
• About the Chi-Square Distribution:
The chi-square distributions are a family
of distributions that take only positive
values and are skewed to the right.
Chi-squared distributions vary depending
on degrees of freedom where df is (n-1)
and n represents the number of
categories for your variable.
Chi-Square Testing
• About the Chi-Square Distribution:
• Each chi-square density curve has the following
properties:
• 1) The total area under each chi-square curve is 1.
• 2) It begins at zero on the horizontal axis, increases to a
peak, and then approached the horizontal axis
asymptotically from above.
• 3) Each curve is skewed to the right. As the number of
degrees of freedom increase, the curve becomes more
and more symmetrical and looks more like a normal
curve (CLT still says this will happen).
Chi-Square Testing
• Chi-square Tests are
*Used with 2-way tables to test for association
or independence with multiple proportions.
*Used to describe relationships within one or
between two categorical variables.
Large 2 values (which equate to low p-values)
are evidence against Ho.
Chi-Square Testing
• Conditions:
• Random: the data comes from a random
sample or a randomized experiment
• Independent: individual observations are
independent or are less than 10% of a large
population
• Large enough sample: all expected counts
are at least 5 (this is essentially the np rules
for each category)
Chi-Square Testing
• There are Three Chi-Square Tests
*Test for Goodness of Fit (GOF)
*Test for Homogeneity
*Test for Independence
Chi-Square (2) test for
Goodness of Fit (GOF)
• Rather than testing individual proportions
in an entire distribution, this test can be
applied to see if the observed sample
distribution is different from the
hypothesized population distribution. (this
is like doing many one proportion z-tests
all at the same time)
Chi-Square GOF Test
Ho: the actual population proportions are equal
to the hypothesized proportions
Ha: at least one of the actual population
proportions is different from the hypothesized
proportions
Chi-Square GOF Test
• The 2 test statistic is:
•
2 = (O – E)2/E
with (n-1) degrees of freedom**
where O – observed value
E – expected value
• ** Remember n is the number of
categories this time, not the sample size
Chi-Square GOF Test
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Calculator Steps for GOF on TI-83 Plus:
1) Clear L1, L2, L3.
2) Enter the observed counts in L1.
3) Calculate expected counts and enter them in L2.
4) Define L3 to be (L1 – L2)2/L2
5) the command Sum(L3) returns the test
statistic 2.
• 6) Use the 2 cdf command from the distributions
menu to ask for the area between your 2 value
and a very large #, and specify the
degrees of freedom. This is your p-value.
Chi-Square GOF Test
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Calculator Steps for GOF on TI-84 Plus:
1) Clear L1, L2, L3.
2) Enter the observed counts in L1.
3) Calculate expected counts and enter them in L2.
4) Stat: Tests: D: 2 GOF
Chi-Square Tests for Homogeneity
• Test for Homogeneity:
Attempts to test determine whether
two populations are similar
(homogeneous) with respect to the
categories of one variable.
Chi-Square Tests for Homogeneity
• Ho: The population proportions with
respect to the variable are the same.
• Ha: The population proportions with
respect to the variable are different.
Chi-Square Test for Independence
• Test for Independence:
One population is sampled and two
characteristics are observed. Is there an
association (dependence) between the
two characteristics.
Chi-Square Test for Independence
• Ho: States there is no association between
the two variables. (The variables are
independent)
• Ha: States that there is an association
between the two variables. (The variables
are not independent)
Chi-Square Tests for Homogeneity
& Independence
• The only difference is in Chi-Square Tests
for Homogeneity & Independence is in
how the data is collected:
• Homogeneity – Two populations
categorized on one categorical variable.
• Independence – One population
categorized on two categorical variables.
Chi-Square Tests for Homogeneity
& Independence
• The test mechanics and everything else
are the same for Homogeneity and
Independence.
• The alternative hypothesis is no longer
one or two sided, it is many-sided. To test
Ho, we compare the observed counts in a
two-way table with the expected counts.
Chi-Square Tests for Homogeneity
& Independence
• Expected cell count = row total x column total
table total
• 2 = (observed – expected)2
expected
• observed are your sample values.
• expected is calculated based on the null.
• In a table with r rows and c columns
df = (r – 1)(c – 1)
Chi-Square Tests for Homogeneity
& Independence
• Calculator instructions:
• 1) Enter the observed in matrix A: 2nd matrix,
edit, choose matrix, enter size & cells
• 2) Stat, tests, C: 2-test, enter name of observed
matrix, enter name of matrix where you
would like the expected to be stored, choose
calculate to compute.