Chi-Square Tests in Statistics
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Categorical data
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1-sample, compared to theoretical distribution
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Goodness-of-Fit Test
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2+ samples, 2+ levels of response variable
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Chi-square Test
Chi-Square Tests
Slide #1
Chi-Square Tests
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Compare observed to theoretical frequencies of
individuals in categories.
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Examples –
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Test whether responses are “random” (e.g., preference)
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Test Mendelian genetics (e.g., 3:1 and 9:3:3:1 theories).
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Test use of available resources (e.g., compare habitat
usage to availability).
Chi-Square Tests
Slide #3
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Determine, at the 10% level, if Northland students
prefer the Chris Duarte Group (CDG), Ronnie
Baker Brooks (RBB), or Bernard Allison (BA).
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Hypotheses?
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H
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: “different # of students prefer each artist”
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H
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: “same # of students prefer each artist”
Chi-Square Tests
Slide #4
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Under H
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If n=78, how many students prefer each artist
if H
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is true?
1/3
26
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Chi-Square Tests
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Suppose these results were obtained:
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Is there a preference – i.e., are these observations
significantly different from what was expected
when assuming no preference?
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Chi-Square Tests
Slide #6
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df = cells - 1
Chi-Square Tests
Slide #7
2
=
2
= 0.15 + 5.54 + 3.85 = 9.54
df
= (3-1) =
2
p-value
= 0.00848
Conclusion?
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Chi-Square Tests
Slide #8
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H
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distribution of individuals into levels follows
the theoretical distribution
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H
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distribution of individuals into levels does
NOT follow the theoretical distribution
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Sample:
randomized, single variable of size n
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Assume:
at least 5 in each cell of
expected table
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Statistic:
Observed frequency table
Chi-Square Tests
Slide #9
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Test Statistic:
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df:
cells-1
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Confidence Region:
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Chi-Square Tests
Slide #10
R - Chi-Square
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A particular type of corn is known to have one of four
types of kernels: purple-smooth, purple-wrinkled, yellow-
smooth, and yellow-wrinkled. The cross between
heterozygous individuals
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should produce a 9:3:3:1 ratio
(in same order of types). Of the kernels on a random cob
32 were purple-smooth
14 were purple-wrinkled
8 were yellow-smooth
4 were yellow-wrinkled
Use the results to determine, at the 5% level, if the
theoretical 9:3:3:1 ratio is upheld with these data.
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i.e., PpSs where t
he purple (P) and smooth (S) alleles are dominant.
Chi-Square Tests
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The leader of a local lakes association conducted a
survey of all members of the association. One question
on the survey was “What is your preferred method of
receiving notices from the lakes association: by regular
mail, by e-mail, by phone, by poster (at the local boat
landing), or other?” Of the surveys returned, 47
respondents preferred regular mail, 63 e-mail, 17
phone, 73 by poster, and 8 some other method. OF THE
RESPONDENTS THAT DID NOT PREFER SOME
OTHER METHOD, is there evidence, at the 5% level,
of a difference in the preferred method of contact?
Chi-Square Tests
Slide #13
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In a randomly selected national sample of 1,007
adults, aged 18 and older, conducted Aug. 22-25,
2005, Gallup polls found that that 403
respondents approved of the way that George W.
Bush was handling his presidency. In a previous
sample (Aug. 8-11, 2005), 45% of the
respondents approved of George W. Bush’s
handling of the presidency. Assuming that this
earlier value was true for the entire population,
determine, at the 5% level, if the approval rating
has changed by the Aug. 22-25, 2005 sample.
Explore the concept of Chi-Square tests through an illustrative example of testing preferences among artists. Learn about Goodness-of-Fit tests, hypotheses, expected versus observed frequencies, new test statistics, and interpreting results through p-values. Discover how these tests compare observed data to theoretical distributions in categorical data analysis.
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Presentation Transcript
Chi-Square Tests Categorical data 1-sample, compared to theoretical distribution Goodness-of-Fit Test 2+ samples, 2+ levels of response variable Chi-square Test Slide #1 Chi-Square Tests
Goodness-of-Fit Test Compare observed to theoretical frequencies of individuals in categories. Examples Test whether responses are random (e.g., preference) Test Mendelian genetics (e.g., 3:1 and 9:3:3:1 theories). Test use of available resources (e.g., compare habitat usage to availability). Slide #2 Chi-Square Tests
An Illustrative Example Determine, at the 10% level, if Northland students prefer the Chris Duarte Group (CDG), Ronnie Baker Brooks (RBB), or Bernard Allison (BA). Hypotheses? Ha: different # of students prefer each artist Ho: same # of students prefer each artist Slide #3 Chi-Square Tests
An Illustrative Example 1/3 Under Ho, what proportion prefer each artist? If n=78, how many students prefer each artist if Ho is true? 26 Artist CDG RBB BA Expected Table Freq 26 26 26 Slide #4 Chi-Square Tests
An Illustrative Example Suppose these results were obtained: Artist CDG RBB BA Observed Table Freq 24 38 16 Is there a preference i.e., are these observations significantly different from what was expected when assuming no preference? Slide #5 Chi-Square Tests
A New Test Statistic ( ) 2 ected observed exp ected table = 2 exp df = cells - 1 Slide #6 Chi-Square Tests
An Illustrative Example Artist CDG RBB BA Observed Table # 24 38 16 Artist CDG RBB BA Expected Table # 26 26 26 ( ) ( )+ ( )+ 2 2 2 26 26 16 26 24 26 38 26 2 = 26 2 = 0.15 + 5.54 + 3.85 = 9.54 df = (3-1) = 2 p-value = 0.00848 Conclusion? Slide #7 Chi-Square Tests
Goodness-of-Fit Test Ho: distribution of individuals into levels follows the theoretical distribution HA: distribution of individuals into levels does NOT follow the theoretical distribution Sample: randomized, single variable of size n Assume: at least 5 in each cell of expected table Statistic: Observed frequency table Slide #8 Chi-Square Tests
Goodness-of-Fit Test ( ) 2 ected observed exp ected table Test Statistic: = 2 exp df: cells-1 ( ) p p 1 *z p Confidence Region: n where is sample proportion in level of interest p Slide #9 Chi-Square Tests
Examine HO Page 5 Slide #10 Chi-Square Tests
