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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.
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)
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
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
Chi-Square Chi-Square: a test of significance used with cross tabulations of nominal/ordinal level data.
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?
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.
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)
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
Are these differences significant? Chi-Square test of significance: • Chi-Square = ∑(fo- fe)2 / fe
Steps Step 1. We have the observed frequencies Political Orientation Liberals Conservatives Child-rearing Permissive 5 10 Not permissive 15 10 ___ ___ 20 20
Steps Step 2. Need to calculate the expected frequencies. Formula: • fe = (row marginal total) (column marginal total) ___________________________________ N
Expected Frequencies • See board
Step 3. Calculate Chi-Square See board
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?
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.
Chi square is sensitive to N (large N’s 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
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
Measure of Association • Which should we use?