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## Making Comparisons

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**Making Comparisons**• All hypothesis testing follows a common logic of comparison • Null hypothesis and alternative hypothesis • mutually exclusive • exhaustive • “Republicans have higher income than Democrats”? • Descriptive, relational, and causal**Experimental Design**• Draw a random sample • Manipulate the independent variable through treatment or intervention • Random assignment into experimental and control groups • Control (keep constant) other outside factors • Observe the effect on the dependent variable**Inferences about Sample Means**• Hypothesis testing is an inferential process • Using limited information to reach a general conclusion • Observable evidence from the sample data • Unobservable fact about the population • Formulate a specific, testable research hypothesis about the population**Null Hypothesis**• no effect, no difference, no change, no relationship, no pattern, no … • any pattern in the sample data is due to random sampling error**Errors in Hypothesis Testing**• Type I Error • A researcher finds evidence for a significant result when, in fact, there is no effect (no relationship) in the population. • The researcher has, by chance, selected an extreme sample that appears to show the existence of an effect when there is none. • The p-value identifies the probability of a Type I error.**Cross-tabulation**• Relationship between two (or more) variables • Joint frequency distribution • Contingency table • Observations should be independent of each other • One person’s response should tell us nothing about another person’s response • Mutually exclusive and exhaustive categories**Cross-tabulation**• If the null hypothesis is true, the independent variable has no effect on the dependent variable • The expected frequency for each cell**Expected Frequency of Each Cell**• Expected frequency in the ith row and the jth column ……… (Eij) • Total counts in the ith row ……… (Ti) • Total counts in the jth column ……… (Tj) • Total counts in the table ……… (N)**Observed frequencies:**• Expected frequencies:**Chi-square (X2)**• For each cell, calculate: • (observed frequency - expected frequency)2 expected frequency • Add up the results from all the cells**Measures of Association**• Symmetrical measures of association • e.g. Kendall’s tau-b and tau-c • Asymmetrical measures of association • e.g. lambda and Somer’s d • Directional measures of association • e.g. Somer’s d • PRE measures of association • e.g. lambda and Somer’s d