Statistical Analyses t-tests Psych 250 Winter, 2013
Dec 24, 2015
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 Type Example Commonly-used Statistical
Method
Nominal by Nominal blood type by gender
Chi-square
Scale by Nominal GPA by gender
GPA by major
t-test
Analysis of Variance
Scale by Scale weight by heightGPA by SAT
RegressionCorrelation
Main Analysis Techniques
Variable Type Example Commonly-used Statistical
Method
Nominal by Nominal blood type by gender
Chi-square
Scale by Nominal GPA by gender
GPA by major
t-test
Analysis of Variance
Scale by Scale weight by heightGPA by SAT
RegressionCorrelation
Main Analysis Techniques
Variables:Scale by Categorical
• Form of the relationship:
Means of each category (M & F victim)
• Statistical Significance:
Independent samples t-test
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
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 monthsF = 11 months
M: 6F: 16
Samples of n = 40 Universe n = ∞
M: 13F: 9
M: 11F: 11
M: 8F: 14
T-test
Sampling distribution: Mean difference
Function of:
1) difference in means
2) variance (dispersion around mean)
Frequency Distributionlengthofsentave11
12 25.0 25.0 25.0
1 2.1 2.1 27.1
1 2.1 2.1 29.2
4 8.3 8.3 37.5
1 2.1 2.1 39.6
4 8.3 8.3 47.9
1 2.1 2.1 50.0
1 2.1 2.1 52.1
8 16.7 16.7 68.8
2 4.2 4.2 72.9
1 2.1 2.1 75.0
4 8.3 8.3 83.3
1 2.1 2.1 85.4
2 4.2 4.2 89.6
2 4.2 4.2 93.8
2 4.2 4.2 97.9
1 2.1 2.1 100.0
48 100.0 100.0
0
1
2
3
4
6
8
10
12
15
16
18
20
24
27
36
60
Total
ValidFrequency Percent Valid Percent
CumulativePercent
Mean = 11
Variance
x i - Mean )2
Variance = s2 = ----------------------- N
x i - Mean )2
but: s2 = ----------------------- N - 1
Standard Deviation = s = variance
Calculating Variancelengthofsentave11
12 25.0 25.0 25.0
1 2.1 2.1 27.1
1 2.1 2.1 29.2
4 8.3 8.3 37.5
1 2.1 2.1 39.6
4 8.3 8.3 47.9
1 2.1 2.1 50.0
1 2.1 2.1 52.1
8 16.7 16.7 68.8
2 4.2 4.2 72.9
1 2.1 2.1 75.0
4 8.3 8.3 83.3
1 2.1 2.1 85.4
2 4.2 4.2 89.6
2 4.2 4.2 93.8
2 4.2 4.2 97.9
1 2.1 2.1 100.0
48 100.0 100.0
0
1
2
3
4
6
8
10
12
15
16
18
20
24
27
36
60
Total
ValidFrequency Percent Valid Percent
CumulativePercent
Mean = 11
Variance
Statistics
lengthofsentave1148
0
11.02
12.109
146.617
0
60
Valid
Missing
N
Mean
Std. Deviation
Variance
Minimum
Maximum
t distribution
• Sampling distribution of a difference in means
• Function of mean difference
& “pooled” variance (of both samples)
mean1 – mean2
t = --------------------------------sp√ (1/n1) + (1/n2)
Null Hyp:M = 11 monthsF = 11 months
mean dif& var
Samples of n = 40 Universe n = ∞
mean dif& var
mean dif& var
mean dif& var
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
Group Statistics
24 16.04 12.723 2.597
24 6.00 9.227 1.883
victim genderfemale
male
lengthofsentave11N Mean Std. Deviation
Std. ErrorMean
Independent Samples Test
.824 .369 3.130 46 .003 10.042 3.208 3.584 16.499
3.130 41.951 .003 10.042 3.208 3.567 16.516
Equal variancesassumed
Equal variancesnot assumed
lengthofsentave11F Sig.
Levene's Test forEquality of Variances
t df Sig. (2-tailed)Mean
DifferenceStd. ErrorDifference Lower Upper
95% ConfidenceInterval of the
Difference
t-test for Equality of Means
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).”
Variable Type Example Commonly-used Statistical Method
Nominal by Nominal blood type by gender
Chi-square
Scale by Nominal GPA by gender
GPA by major
t-test
Analysis of Variance
Scale by Scale weight by heightGPA by SAT
RegressionCorrelation
Analysis of Variance
Dep Var: Length of SentenceIndep var: Major
length of sentence
12 25.0 25.0 25.0
1 2.1 2.1 27.1
4 8.3 8.3 35.4
5 10.4 10.4 45.8
1 2.1 2.1 47.9
1 2.1 2.1 50.0
6 12.5 12.5 62.5
2 4.2 4.2 66.7
6 12.5 12.5 79.2
1 2.1 2.1 81.3
1 2.1 2.1 83.3
2 4.2 4.2 87.5
1 2.1 2.1 89.6
1 2.1 2.1 91.7
1 2.1 2.1 93.8
1 2.1 2.1 95.8
1 2.1 2.1 97.9
1 2.1 2.1 100.0
48 100.0 100.0
0
1
2
3
4
5
6
8
12
15
16
18
24
27
36
42
60
66
Total
ValidFrequency Percent Valid Percent
CumulativePercent
Mean = 14.6
Variance = 212.4
Statistics
length of sentence48
0
9.98
14.573
212.361
Valid
Missing
N
Mean
Std. Deviation
Variance
Length of Sentence by Major
• Nat sci 14.3
• Soc sci 7.4
• Art & Hum 11.0
Descriptives
lengthofsentave11
19 14.26 15.183 3.483 6.94 21.58 0 60
14 7.43 8.474 2.265 2.54 12.32 0 24
15 10.27 10.067 2.599 4.69 15.84 0 36
48 11.02 12.109 1.748 7.50 14.54 0 60
natural science
social science
arts and humanities
Total
N Mean Std. Deviation Std. Error Lower Bound Upper Bound
95% Confidence Interval forMean
Minimum Maximum
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:
nsocsci(Meansocsci - Mean)2
+ narthum(Meanarthum - Mean)2
+nnatsci(Meannatsci – Mean)2 / df
• Within groups:
(ni – Meansocsci) 2
+ (ni - Meanarthum)2
+ (ni - Meannatsci) 2 / df
Null Hyp:Nat sci = 11 monthsSoc sci = 11 monthsArt-Hum = 11 months
f
Samples of n = 40 Universe n = ∞
f
f
f
ANOVA: sentence by major
Descriptives
lengthofsentave11
19 14.26 15.183 3.483 6.94 21.58 0 60
14 7.43 8.474 2.265 2.54 12.32 0 24
15 10.27 10.067 2.599 4.69 15.84 0 36
48 11.02 12.109 1.748 7.50 14.54 0 60
natural science
social science
arts and humanities
Total
N Mean Std. Deviation Std. Error Lower Bound Upper Bound
95% Confidence Interval forMean
Minimum Maximum
ANOVA
lengthofsentave11
388.933 2 194.467 1.346 .271
6502.046 45 144.490
6890.979 47
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
ANOVA: sentence by majorsimulated data
Descriptives
lengthofsentave11
19 14.26 15.183 3.483 6.94 21.58 0 60
14 7.43 8.474 2.265 2.54 12.32 0 24
15 10.27 10.067 2.599 4.69 15.84 0 36
48 11.02 12.109 1.748 7.50 14.54 0 60
natural science
social science
arts and humanities
Total
N Mean Std. Deviation Std. Error Lower Bound Upper Bound
95% Confidence Interval forMean
Minimum Maximum
ANOVA
lengthofsentave11
388.933 2 194.467 1.346 .271
6502.046 45 144.490
6890.979 47
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
ANOVA: sentence by majorsimulated data
Descriptives
lengthofsentave11
19 14.26 15.183 3.483 6.94 21.58 0 60
14 7.43 8.474 2.265 2.54 12.32 0 24
15 10.27 10.067 2.599 4.69 15.84 0 36
48 11.02 12.109 1.748 7.50 14.54 0 60
natural science
social science
arts and humanities
Total
N Mean Std. Deviation Std. Error Lower Bound Upper Bound
95% Confidence Interval forMean
Minimum Maximum
ANOVA
lengthofsentave11
388.933 2 194.467 1.346 .271
6502.046 45 144.490
6890.979 47
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.