DICE!
 You
are going to make
your own and then we
are going to test them
(later) to see if they are
fair! 
Chapter 11
Chi-Squared 
(Categorical Data)
Chi-Squared
 The
chi-squared goodness of fit test allows
us to determine whether a hypothesized
distribution seems valid.
 (multiple variables in a distribution – not
just one)
 Two types:


Chi-squared for homogeneity – tells us
whether the distributions differ when there is
a treatment/experiment involved.
Chi-squared for association – tells us
whether the distributions differ in an
observational study.
Chi-Squared Statistic
 𝑥2
=
(𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑 −𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑)2
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑
Carrying Out a Test
 Conditions
– Random, Large Sample Size (all
counts are at least 5), Independent (individual
observations are independent. When sampling
without replacement, 10% of population rule)
 𝐻0: The specified distribution is correct
 𝐻𝑎: The specified distribution is not correct
 𝑥2
=
 Find
(𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑 −𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑)2
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑
p-value with df=k-1 (number of categories -1)
 The
chi-squared test statistic compares
observed minus expected COUNTS. Don’t
try to perform calculations with observed
minus expected proportions in each
category!
 The checking large sample size condition,
be sure to examine the EXPECTED counts,
not the observed counts.
When were you born?
 Are
births evenly distributed across the
days of the week? The one-way table
shows results of a random sample of 140
births from local records.
Days
Sun
Mon
Tues
Wed
Thurs
Fri
Sat
Births
13
23
24
20
27
18
15
 Do
these data give significant evidence
that local births are not equally likely on all
days of the week?
 State:
 Plan:
 Do:
 Conclude:
Follow-Up Analysis
 When
you reject your null hypothesis…you
need to follow up by looking at individual
components to see which values are the
biggest contributors – or helped to push
your data far enough to reject.
 These components show which terms
contribute most to the chi-squared
statistic.
Homework
 Pg
692 (1-9 odd, 11, 15, 19-22)
 Let’s do #11 together! 
#11