Chi-square test or ?2 test

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Chi-square testor?2 test
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What if we are interested in seeing if my crazy
dice are considered fair? What can I do?
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Chi-square test

• Used to test the counts of categorical data
• Three types
• Goodness of fit (univariate)
• Independence (bivariate)
• Homogeneity (univariate with two samples)

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?2 distribution
df3
df5
df10
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?2 distribution

• Different df have different curves
• Skewed right
• As df increases, curve shifts toward right
  becomes more like a normal curve

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?2 assumptions

• SRS reasonably random sample
• Have counts of categorical data we expect each
  category to happen at least once
• Sample size to insure that the sample size is
  large enough we should expect at least five in
  each category.
• Be sure to list expected counts!!

Combine these together All expected counts are
at least 5.
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?2 formula
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?2 Goodness of fit test
Based on df df number of categories - 1

• Uses univariate data
• Want to see how well the observed counts fit
  what we expect the counts to be
• Use ?2cdf function on the calculator to find
  p-values

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Hypotheses written in words

• H0 the observed counts equal the expected
  counts
• Ha the observed counts are not equal to
  the expected counts
• Be sure to write in context!

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Lets test our dice!
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Does your zodiac sign determine how successful
you will be? Fortune magazine collected the
zodiac signs of 256 heads of the largest 400
companies. Is there sufficient evidence to claim
that successful people are more likely to be born
under some signs than others? Aries
23 Libra 18 Leo 20 Taurus 20 Scorpio 21 Virgo
19 Gemini 18 Sagittarius 19 Aquarius 24 Cancer 2
3 Capricorn 22 Pisces 29 How many would you
expect in each sign if there were no difference
between them? How many degrees of freedom?
I would expect CEOs to be equally born under all
signs. So 256/12 21.333333
Since there are 12 signs df 12 1 11
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• Assumptions
• Have a random sample of CEOs
• All expected counts are greater than 5. (I expect
  21.33 CEOs to be born in each sign.)
• H0 The number of CEOs born under each sign is
  the same.
• Ha The number of CEOs born under each sign is
  different.
• P-value c2cdf(5.094, 1099, 11) .9265 a
  .05
• Since p-value gt a, I fail to reject H0. There is
  not sufficient evidence to suggest that the CEOs
  are born under some signs than others.

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A company says its premium mixture of nuts
contains 10 Brazil nuts, 20 cashews, 20
almonds, 10 hazelnuts and 40 peanuts. You buy
a large can and separate the nuts. Upon weighing
them, you find there are 112 g Brazil nuts, 183 g
of cashews, 207 g of almonds, 71 g or hazelnuts,
and 446 g of peanuts. You wonder whether your
mix is significantly different from what the
company advertises? Why is the chi-square
goodness-of-fit test NOT appropriate here? What
might you do instead of weighing the nuts in
order to use chi-square?
Because we do NOT have counts of the type of nuts.
We could count the number of each type of nut and
then perform a c2 test.
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Offspring of certain fruit flies may have yellow
or ebony bodies and normal wings or short wings.
Genetic theory predicts that these traits will
appear in the ratio 9331 (yellow normal,
yellow short, ebony normal, ebony short) A
researcher checks 100 such flies and finds the
distribution of traits to be 59, 20, 11, and 10,
respectively. What are the expected counts?
df? Are the results consistent with the
theoretical distribution predicted by the genetic
model? (see next page)
Since there are 4 categories, df 4 1 3
Expected counts Y N 56.25 Y S 18.75 E
N 18.75 E S 6.25
We expect 9/16 of the 100 flies to have yellow
and normal wings. (Y N)
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• Assumptions
• Have a random sample of fruit flies
• All expected counts are greater than 5.
• Expected counts
• Y N 56.25, Y S 18.75, E N 18.75, E
  S 6.25
• H0 The distribution of fruit flies is the same
  as the theoretical model.
• Ha The distribution of fruit flies is not the
  same as the theoretical model.
• P-value c2cdf(5.671, 1099, 3) .129 a .05
• Since p-value gt a, I fail to reject H0. There is
  not sufficient evidence to suggest that the
  distribution of fruit flies is not the same as
  the theoretical model.

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?2 test for independence

• Used with categorical, bivariate data from ONE
  sample
• Used to see if the two categorical variables are
  associated (dependent) or not associated
  (independent)

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Assumptions formula remain the same!
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Hypotheses written in words

• H0 two variables are independent
• Ha two variables are dependent
• Be sure to write in context!

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A beef distributor wishes to determine whether
there is a relationship between geographic region
and cut of meat preferred. If there is no
relationship, we will say that beef preference is
independent of geographic region. Suppose that,
in a random sample of 500 customers, 300 are from
the North and 200 from the South. Also, 150
prefer cut A, 275 prefer cut B, and 75 prefer cut
C.
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If beef preference is independent of geographic
region, how would we expect this table to be
filled in?
North South Total
Cut A 150
Cut B 275
Cut C 75
Total 300 200 500
90
60
165
110
45
30
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Expected Counts

• Assuming H0 is true,

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Degrees of freedom
Or cover up one row one column count the
number of cells remaining!
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Now suppose that in the actual sample of 500
consumers the observed numbers were as
follows   (on your paper)  Is there sufficient
evidence to suggest that geographic regions and
beef preference are not independent? (Is there a
difference between the expected and observed
counts?)
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• Assumptions
• Have a random sample of people
• All expected counts are greater than 5.
• H0 geographic region and beef preference are
  independent Ha geographic region and beef
  preference are dependent
• P-value .0226 df 2 a .05
• Since p-value lt a, I reject H0. There is
  sufficient evidence to suggest that geographic
  region and beef preference are dependent.

Expected Counts N S A 90
60 B 165 110 C 45 30
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?2 test for homogeneity

• Used with a single categorical variable from two
  (or more) independent samples
• Used to see if the two populations are the same
  (homogeneous)

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Assumptions formula remain the same! Expected
counts df are found the same way as test for
independence. Only change is the hypotheses!
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Hypotheses written in words

• H0 the two (or more) distributions are the same
• Ha the distributions are different
• Be sure to write in context!

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The following data is on drinking behavior for
independently chosen random samples of male and
female students. Does there appear to be a
gender difference with respect to drinking
behavior? (Note low 1-7 drinks/wk, moderate
8-24 drinks/wk, high 25 or more drinks/wk)
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Expected Counts M
F 0 158.6 167.4 L 554.0 585.0 M 230.1 243.0 H 38.4
40.6

• Assumptions
• Have 2 random sample of students
• All expected counts are greater than 5.
• H0 drinking behavior is the same for female
  male students Ha drinking behavior is not
  the same for female male students
• P-value .000 df 3 a .05
• Since p-value lt a, I reject H0. There is
  sufficient evidence to suggest that drinking
  behavior is not the same for female male
  students.