Inferential Statistics

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Inferential Statistics

• Hypothesis testing (relationship between 2 or
  more variables)
• We want to make inferences from a sample to a
  population.
• A random sample allows us to infer from a sample
  to a population.

2
Inferential Statistics

• Significance Tests
• Z scores (one sample case)
• Difference of means tests
• Two sample case (t-test)
• Three or more sample case (ANOVA)
• Chi-Square
• Bi-Variate Correlation (One IV One DV)
• Bi-Variate Regression (One IV One DV)
• Multi-Variate Regression (Two or more IVs One
  DV)

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Level of Measurement Significance Tests

• Chi-Square
• IV DV are nominal and/or ordinal
• t-test
• IV is nominal (group like men women)
• DV is Interval/Ratio (or a scale)
• ANOVA
• IV is nominal (group with 3 or more categories)
• DV is I/R (or a scale)
• Regression
• IV(s) DV are I/R (or scales)
• IV(s) can be dummy variables

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Which Test Would you Use?

• Hr There is a relationship between
• gender income (measured in dollars)
• race (measured as Black, Latino/a,
  Caucasian) and income
• religious preference (catholic, protestant)
  and attitudes toward abortion (favor, oppose)
• education (measured in years) and income
• degree completed (HS or Less College) and
  income

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Chi-Square

• Chi-Square a test of significance used with
  cross tabulations of nominal/ordinal level data.

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• Example
• Research question Does political orientation
  influence parenting style?
• Political orientation Conservative Liberal
• Parenting style Permissive Not Permissive
• Why not simply compare the mean difference
• between liberals and conservatives on parenting
• style?

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• We are really saying
• Hr The frequency (proportion) of liberals who
  are permissive is not the same as the frequency
  of conservatives who are permissive.
• The null (a hypothesis of no difference) says
• Ho The frequency (proportion) of liberals who
  are permissive is the same as the frequency of
  conservatives who are permissive.

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• Chi-Square compares the observed frequencies
  (from the data in your sample) to expected
  frequencies.
• Expected frequencies These are the frequencies
  we would expect if the null were true (if there
  is no difference between political view and
  parenting style)

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• Example
• We do a cross tab of political orientation by
  parenting style and our observed frequencies are
• Political Orientation
• Liberals Conservatives
• Child-rearing
• Permissive 5 10
• Not permissive 15 10
• ___ ___
• 20 20

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• Are these differences significant?
• Chi-Square test of significance
• Chi-Square ?(fo- fe)2 / fe

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Steps

• Step 1. We have the observed frequencies
• Political Orientation
• Liberals Conservatives
• Child-rearing
• Permissive 5 10
• Not permissive 15 10
• ___ ___
• 20 20

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Steps

• Step 2. Need to calculate the expected
  frequencies.
• Formula
• fe (row marginal total) (column marginal total)
• ___________________________________
• N

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Expected Frequencies

• See board

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• Step 3. Calculate Chi-Square
• See board

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• Calculated Chi-Square for Political Views by
  Parenting Style
• Chi Square 2.66
• Df (r-1)(c-1)
• Df (2-1) (2-1) 1
• Must have a Chi Square of 3.84 at p..05
• to reject the null hypothesis.
• Decision?

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Review Alpha Levels

• Alpha level the probability of making a Type I
  error
• Type I error (reject the null when it is true)
• Set alpha level small (.05 or smaller) to
  minimize risk.
• The larger the sample the smaller the alpha level
  should be.

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• Chi square is sensitive to N (large Ns can yield
  significant results)
• So, we use a measure of association with
  Chi-square
• Measures of association tell us about the
• strength of the relationship

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• Measures of Association
• The type of measure used is determined by the
  level of measurement and the number of
  categories.
• See handout
• Interpret GSS Output

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Crosstab
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Chi-Square
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Measure of Association

• Which should we use?

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• Cramers V .112