Political Science 5: Example of ChiSquared Test

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Political Science 5Example of Chi-Squared Test
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How Sure is Sure? Quantifying Uncertainty in
Tables

• Using Two-Way Tables
• SAT scores and UC admissions
• Null hypothesis there is no relationship between
  race/ethnicity and admissions
• Chi-Square Test
• The formula
• Interpreting your results
• Using a Chi-Square test to hold constant a
  confound or another independent variable

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Race and Admissions

• A recent study of UC admissions over the past two
  years showed that thousands of students with SATs
  under 1000 have been admitted.
• Even flagships like UCSD, UCLA and Berkeley
  admitted hundreds.
• Many observers offered this hypothesis to explain
  the trend Minority students with low SATs are
  more likely to be admitted.

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Race and Admissions

• Lets examine the bivariate correlation between
  ethnicity and admittance.

Data on UC applicants who scored lower than 1000
on the SAT, taken from the Los Angeles Times,
November 3, 2003.
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Race and Admissions

• If the ethnicity of a student with sub-1000 SAT
  scores makes him more likely to be admitted to
  the UC, admission rates in that table will differ
  by race.
• 62.9 for underrepresented minorities
• 63.4 for whites and Asian-Americans
• Opposite of what the hypothesis predicted!

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Test of Statistical Significance

• We still want to see if this small difference in
  our sample is statistically significant at the
  95 confidence level.
• We could do a difference in proportions test.
• We will do a Chi-Square test, a nice option when
  we have more than two groups.
• The null hypothesis is that there is no
  association between ethnicity and UC admission
  rates for these students.

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Chi-Square Test

• Under the null hypothesis, you would expect to
  see table entries that indicate the same
  admission rates for each ethnicity.
• To calculate the expected count in any cell of a
  two-way table when the null hypothesis is true

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Chi-Square Test

• For the Row 1, Column 1 entry in our UC
  admission table, which gives the number of
  underrepresented minorities who are admitted,
  this expected count is

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Chi-Square Test

• To conduct a Chi-Square test, first fill in a
  tables worth of expected counts

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Chi-Square Test

• Then use this formula to get a single Chi-Square
  statistic, which is a measure of how much the
  observed counts differ from the expected counts,
  for your table

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Chi-Square Test

• So for each entry
• Step 1. Subtract the expected count from the
  observed count.
• Step 2. Square that difference.
• Step 3. Divide that square by the expected count
  to get a quotient.
• Then add up your quotients for all of the cells.

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Chi-Square Test

• When the null hypothesis is true, the resulting
  Chi-Square statistic has a sampling distribution
  called the chi-square distribution,
  characterized by its degrees of freedom, which
  equals
• df ( of rows - 1) ( of columns - 1)
• Our UC admissions table has 1 degree of freedom
• (2-1) (2-1) 1 1 1

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Chi-Square Test

• We can reject the null hypothesis at the 95
  confidence level if our Chi-Square statistic is
  greater than the values given by this table for
  each number of degrees of freedom.
• Chi-Square is 0.76 in UC example, so cannot reject

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Chi-Square Test

• If you have a confound, like Did the students
  parent attend UC?, you can use the Chi-Square
  test to see if there is a relationship between
  ethnicity and admission, holding confound
  constant
• Build one 2X2 table for students who parents
  attended a UC, and another 2X2 table for students
  with no legacy
• Conduct two separate Chi-Square tests to examine
  bivariate relationship for each group.