Understanding Chi-Square Tests in Statistics
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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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
