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Chapter 11 - Review Quantitative Data Analyses so far One...

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Review: Quantitative Data Analyses so far
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σ 2 =? p = ? µ = ? σ known σ not known z -test t -test z test chi square test One population tests µ 1 = µ 2 Dependent Paired Blocked (2trts) paired t -test Independent Unpaired CRD (2trts) 2 sample t -test 2 populations test
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Quantitative Bivariate Data Correlation Regression ρ = 0 β 1 = 0 t-test for correlation t-test for regression What about Qualitative Data?
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Chapter 11 Applications of Chi Square
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Chi-Square Statistic Qualitative Data Have Observed numbers = Counts Ex: The number of men vs. women that smoke or don’t smoke Three Types of Chi Square Tests 1) Goodness of Fit Tests 2) Tests of Independence 3) Tests of Homogeneity
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All 3 tests use same statistic equation Different equation (from Chapter 5) but same Table 8 O = Observed Number E = Expected Number 1 tailed tests: interest is the right tail = big χ 2 * values What does a small χ 2 * value mean? - = E E O 2 2 ) ( * χ
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- = E E O 2 2 ) ( * χ 0 0
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Chi Square Tests 1) Goodness of Fit Tests Book calls multinomial experiments 2) Tests of Independence 3) Tests of Homogeneity (treatment effects) Qualitative Data (observed #s) 1 qualitative variable GOF test 2 qualitative variables Test of Independence Test of Homogeneity
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Goodness of Fit Tests Flashback: Do data fit binomial distribution? Key Points: You have a single list of observed numbers in categories Question drives hypotheses H o : Data follow expected distribution H a : Data don’t follow expected distribution. Hypotheses often verbal statements
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Do car crashes occur on different days with the same frequency? Sample from Maryland in May 2004 Example Test the claim that accidents occur with equal frequency Use α = 0.05 What is the expected # for each square given the question ? Day Sun Mon Tue Wed Thur Fri Sat # Fatalities 31 20 20 22 22 29 36
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1 . State your question (ID population) Are fatal auto accidents equally likely on all days in Maryland? 2. State your H o and H a (define symbols) H o : fatal auto accidents in Maryland are equally likely on all days H a : fatal auto accidents in Maryland are not equally likely on all days
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Calculate test statistic need to calculate expected values expected #s based on null hypothesis here: accidents equally likely on all days so: take the total # of accidents and divide by 7 to get the expected # for each day =180 ∕ 7 = 25.7 -Chi Square tests do have 1 Assumption: All EXPECTED values > 5 25.7 bigger than 5 so that assumption okay
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Day Sun Mon Tue Wed Thu Fri Sat # fatalities Observed 31 20 20 22 22 29 36 # fatalities Expected 25.7 25.7 25.7 25.7 25.7 25.7 25.7 - = E E O 2 2 ) ( * χ
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χ 2* = (31-25.7) 2 / 25.7 + (20-25.7) 2 /25.7 + (20-25.7) 2 / 25.7 + (22-25.7)
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