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Statistical Analyses t-tests Psych 250 Winter, 2013
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Hypothesis: People will give longer sentences when the victim is female.
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Independent Variable: Gender of the Victim Dependent Variable: Length of Sentence
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Types of Measures / Variables Nominal / categorical –Gender, major, blood type, eye color Ordinal –Rank-order of favorite films; Likert scales? Interval / scale –Time, money, age, GPA
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Variable TypeExampleCommonly-used Statistical Method Nominal by Nominalblood type by genderChi-square Scale by NominalGPA by gender GPA by major t-test Analysis of Variance Scale by Scaleweight by height GPA by SAT Regression Correlation Main Analysis Techniques
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Variable TypeExampleCommonly-used Statistical Method Nominal by Nominalblood type by genderChi-square Scale by NominalGPA by gender GPA by major t-test Analysis of Variance Scale by Scaleweight by height GPA by SAT Regression Correlation Main Analysis Techniques
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Stat Analysis / Hypothesis Testing 1.Form of the relationship 2.Statistical significance
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Variables: Scale by Categorical Form of the relationship: Means of each category (M & F victim) Statistical Significance: Independent samples t-test
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Means observed in Sample Victim GenderAverage Sentence Male6 months Female16 months
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Statistical Signficance Q: Is this a “statistically significant” difference? Can the “null hypothesis” be rejected? Null hypothesis: there are NO differences in sentencing for male vs. female victims
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Universe n = ∞ Sample n = 40 M victim: 6 months F victim: 16 months sample inference
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Logic of Statistical Inference What is the probability of drawing the observed sample (M = 6 months vs. F = 16 months) from a universe with no differences? If probability very low, then differences in sample likely reflect differences in universe Then null hypothesis can be rejected; difference in sample is statistically significant
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Strategy Draw an infinite number of samples of n = 40, and graph the distribution of their male victim / female victim differences
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Null Hyp: M = 11 months F = 11 months M: 6 F: 16 Samples of n = 40 Universe n = ∞ M: 13 F: 9 M: 11 F: 11 M: 8 F: 14
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T-test Sampling distribution: Mean difference Function of: 1) difference in means 2) variance (dispersion around mean)
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Possible Sample -- 1 1 2 3 4 5 6... 16 Male Victim Female Victim
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Possible Sample -- 2 1 2 3 4 5 6... 16 Male Victim Female Victim
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Frequency Distribution Mean = 11
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Variance x i - Mean ) 2 Variance = s 2 = ----------------------- N x i - Mean ) 2 but:s 2 = ----------------------- N - 1 Standard Deviation = s = variance
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Calculating Variance Mean = 11
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Variance
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t distribution Sampling distribution of a difference in means Function of mean difference & “pooled” variance (of both samples) mean 1 – mean 2 t = -------------------------------- s p √ (1/n 1 ) + (1/n 2 )
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Null Hyp: M = 11 months F = 11 months mean dif & var Samples of n = 40 Universe n = ∞ mean dif & var mean dif & var mean dif & var
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Null Hyp: M = 11 months F = 11 months t Samples of n = 40 Universe n = ∞ t t t
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t distribution 2.5% of area
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Statistical Significance If probability is less than 5 in 100, the null hypothesis can be rejected, and it can be concluded that the difference also exists in the universe. p <.05 The finding from the sample is statistically significant
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SPSS t-test Output 1. Read means 2. Read Levene’s Test 3. Read p value
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Report Findings “Assailants were given an average sentence of 16 months when the victims were female, compared to 6 months when the victims were male (df = 46, t = 3.13, p. <.005).” “Respondents gave longer sentences when the victims were female (16 months) than when they were male (6 months), a difference that was statistically signficant (df = 46, t = 3.13, p. <.005).”
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Statistical Analyses analysis of variance ( ANOVA ) Psych 250 Winter, 2011
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Variable TypeExampleCommonly-used Statistical Method Nominal by Nominalblood type by gender Chi-square Scale by NominalGPA by gender GPA by major t-test Analysis of Variance Scale by Scaleweight by height GPA by SAT Regression Correlation Analysis of Variance
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Dep Var: Length of Sentence Indep var: Major Mean = 14.6 Variance = 212.4
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Form of Relationship (differences seen in sample)
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Length of Sentence by Major Nat sci14.3 Soc sci 7.4 Art & Hum11.0
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Statistical Inference ( generalize from sample to universe? )
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Universe n = ∞ Sample n = 40 Nat sci = 14.3 Soc sci = 7.4 A & H = 11.0 sample inference
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Possible Sample -- 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Social Science Art & Human Natural Science
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Possible Sample -- 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Social Science Art & Human Natural Science
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ANOVA Logic 1.Calculate ratio of “between-groups” variance to “within-groups” variance 2.Estimate the sampling distribution of that ratio:F distribution 3.If the probability that the ratio in sample could come from universe with no differences in group means is <.05, can reject null hypothesis and infer that mean differences exist in universe
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ANOVA Logic Between groups: n socsci (Mean socsci - Mean) 2 + n arthum (Mean arthum - Mean) 2 +n natsci (Mean natsci – Mean) 2 / df Within groups: (n i – Mean socsci ) 2 + (n i - Mean arthum ) 2 + (n i - Mean natsci ) 2 / df
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F ratio between groups mean squares F = within groups mean squares
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Null Hyp: Nat sci = 11 months Soc sci = 11 months Art-Hum = 11 months f Samples of n = 40 Universe n = ∞ f f f
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f Distributions
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ANOVA: sentence by major
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ANOVA: sentence by major simulated data
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Write Findings “Social science majors assigned sentences averaging 7.4 years, arts and humanities students 10.3 years, and natural science students 14.3 years, but these differences were not statistically significant (df = 2, 42, F = 1.35, p <.30).”
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