Statistical Analyses t-tests Psych 250 Winter, 2013

Hypothesis: People will give longer sentences when the victim is female.

Independent Variable: Gender of the Victim Dependent Variable: Length of Sentence

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

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

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

Stat Analysis / Hypothesis Testing 1.Form of the relationship 2.Statistical significance

Variables: Scale by Categorical Form of the relationship: Means of each category (M & F victim) Statistical Significance: Independent samples t-test

Means observed in Sample Victim GenderAverage Sentence Male6 months Female16 months

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

Universe n = ∞ Sample n = 40 M victim: 6 months F victim: 16 months sample inference

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

Strategy Draw an infinite number of samples of n = 40, and graph the distribution of their male victim / female victim differences

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

T-test Sampling distribution: Mean difference Function of: 1) difference in means 2) variance (dispersion around mean)

Possible Sample Male Victim Female Victim

Possible Sample Male Victim Female Victim

Frequency Distribution Mean = 11

Variance   x i - Mean ) 2 Variance = s 2 = N   x i - Mean ) 2 but:s 2 = N - 1 Standard Deviation = s =  variance

Calculating Variance Mean = 11

Variance

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 )

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

Null Hyp: M = 11 months F = 11 months t Samples of n = 40 Universe n = ∞ t t t

t distribution 2.5% of area

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

SPSS t-test Output 1. Read means 2. Read Levene’s Test 3. Read p value

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).”

Statistical Analyses analysis of variance ( ANOVA ) Psych 250 Winter, 2011

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

Dep Var: Length of Sentence Indep var: Major Mean = 14.6 Variance = 212.4

Form of Relationship (differences seen in sample)

Length of Sentence by Major Nat sci14.3 Soc sci 7.4 Art & Hum11.0

Statistical Inference ( generalize from sample to universe? )

Universe n = ∞ Sample n = 40 Nat sci = 14.3 Soc sci = 7.4 A & H = 11.0 sample inference

Possible Sample Social Science Art & Human Natural Science

Possible Sample Social Science Art & Human Natural Science

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

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

F ratio between groups mean squares F = within groups mean squares

Null Hyp: Nat sci = 11 months Soc sci = 11 months Art-Hum = 11 months f Samples of n = 40 Universe n = ∞ f f f

f Distributions

ANOVA: sentence by major

ANOVA: sentence by major simulated data

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).”