Download Statistical Analyses

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
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
t-test
GPA by major
Analysis of Variance
weight by height
GPA by SAT
Regression
Correlation
Scale by Scale
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
t-test
GPA by major
Analysis of Variance
weight by height
GPA by SAT
Regression
Correlation
Scale by Scale
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 Gender
Average Sentence
Male
6 months
Female
16 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
sample
Sample
n = 40
inference
M victim: 6 months
F victim: 16 months
Universe
n=∞
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
Samples of n = 40
Universe n = ∞
M: 13
F: 9
M: 6
F: 16
Null Hyp:
M = 11 months
F = 11 months
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 -- 1
Male Victim
1
2
3
4
5
6
...
Female Victim
16
Possible Sample -- 2
Male Victim
1
2
3
4
5
6
...
Female Victim
16
Frequency Distribution
lengthofsentave11
Valid
0
1
2
3
4
6
8
10
12
15
16
18
20
24
27
36
60
Total
Frequency
12
1
1
4
1
4
1
1
8
2
1
4
1
2
2
2
1
48
Percent
25.0
2.1
2.1
8.3
2.1
8.3
2.1
2.1
16.7
4.2
2.1
8.3
2.1
4.2
4.2
4.2
2.1
100.0
Valid Percent
25.0
2.1
2.1
8.3
2.1
8.3
2.1
2.1
16.7
4.2
2.1
8.3
2.1
4.2
4.2
4.2
2.1
100.0
Cumulative
Percent
25.0
27.1
29.2
37.5
39.6
47.9
50.0
52.1
68.8
72.9
75.0
83.3
85.4
89.6
93.8
97.9
100.0
Mean = 11
Variance
Variance = s2 =
but:
  x i - Mean )2
----------------------N
s2 =
Standard Deviation =
  x i - Mean )2
----------------------N-1
s
=  variance
Calculating Variance
lengthofsentave11
Valid
0
1
2
3
4
6
8
10
12
15
16
18
20
24
27
36
60
Total
Frequency
12
1
1
4
1
4
1
1
8
2
1
4
1
2
2
2
1
48
Percent
25.0
2.1
2.1
8.3
2.1
8.3
2.1
2.1
16.7
4.2
2.1
8.3
2.1
4.2
4.2
4.2
2.1
100.0
Valid Percent
25.0
2.1
2.1
8.3
2.1
8.3
2.1
2.1
16.7
4.2
2.1
8.3
2.1
4.2
4.2
4.2
2.1
100.0
Cumulative
Percent
25.0
27.1
29.2
37.5
39.6
47.9
50.0
52.1
68.8
72.9
75.0
83.3
85.4
89.6
93.8
97.9
100.0
Mean = 11
Variance
Statistics
lengthofsentave11
N
Valid
Mis sing
Mean
Std. Deviation
Variance
Minimum
Maximum
48
0
11.02
12.109
146.617
0
60
t distribution
• Sampling distribution of a difference in
means
• Function of mean difference
& “pooled” variance (of both samples)
t =
mean1 – mean2
-------------------------------sp√ (1/n1) + (1/n2)
Samples of n = 40
Universe n = ∞
mean dif
& var
mean dif
& var
Null Hyp:
M = 11 months
F = 11 months
mean dif
& var
mean dif
& var
Samples of n = 40
Universe n = ∞
t
t
Null Hyp:
M = 11 months
F = 11 months
t
t
t distribution
2.5% of area
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
Group Statistics
lengthofsentave11
victim gender
female
male
N
Mean
16.04
6.00
24
24
Std. Deviation
12.723
9.227
Std. Error
Mean
2.597
1.883
Independent Samples Test
Levene's Test for
Equality of Variances
F
lengthofsentave11
Equal variances
ass umed
Equal variances
not as sumed
.824
Sig.
.369
2. Read Levene’s Test
t-tes t for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
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
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
Analysis of Variance
Variable Type
Example
Commonly-used
Statistical Method
Nominal by Nominal
blood type by
gender
Chi-square
Scale by Nominal
GPA by gender
t-test
GPA by major
Analysis of Variance
weight by height
GPA by SAT
Regression
Correlation
Scale by Scale
Dep Var: Length of Sentence
Indep var: Major
length of sentence
Valid
0
1
2
3
4
5
6
8
12
15
16
18
24
27
36
42
60
66
Total
Frequency
12
1
4
5
1
1
6
2
6
1
1
2
1
1
1
1
1
1
48
Percent
25.0
2.1
8.3
10.4
2.1
2.1
12.5
4.2
12.5
2.1
2.1
4.2
2.1
2.1
2.1
2.1
2.1
2.1
100.0
Valid Percent
25.0
2.1
8.3
10.4
2.1
2.1
12.5
4.2
12.5
2.1
2.1
4.2
2.1
2.1
2.1
2.1
2.1
2.1
100.0
Statistics
Cumulative
Percent
25.0
27.1
35.4
45.8
47.9
50.0
62.5
66.7
79.2
81.3
83.3
87.5
89.6
91.7
93.8
95.8
97.9
100.0
length of s entence
N
Valid
Mis sing
Mean
Std. Deviation
Variance
48
0
9.98
14.573
212.361
Mean = 14.6
Variance = 212.4
Form of Relationship
(differences seen in sample)
Length of Sentence by Major
Descriptives
lengthofsentave11
N
natural s cience
s ocial s cience
arts and humanities
Total
19
14
15
48
Mean
14.26
7.43
10.27
11.02
Std. Deviation
15.183
8.474
10.067
12.109
Std. Error
3.483
2.265
2.599
1.748
• Nat sci
• Soc sci
• Art & Hum
95% Confidence Interval for
Mean
Lower Bound Upper Bound
6.94
21.58
2.54
12.32
4.69
15.84
7.50
14.54
14.3
7.4
11.0
Minimum
0
0
0
0
Maximum
60
24
36
60
Statistical Inference
( generalize from sample to
universe? )
sample
Sample
n = 40
inference
Nat sci = 14.3
Soc sci = 7.4
A & H = 11.0
Universe
n=∞
Possible Sample -- 1
Social Science
1
2
3
4
5
6
7
8
Art & Human Natural Science
9
10
11
12
13
14
15
Possible Sample -- 2
Social Science
1
2
3
4
5
6
7
8
Art & Human Natural Science
9
10
11
12
13
14
15
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
F ratio
between groups mean squares
F =
within groups mean squares
Samples of n = 40
Universe n = ∞
f
f
Null Hyp:
Nat sci = 11 months
Soc sci = 11 months
Art-Hum = 11 months
f
f
f Distributions
ANOVA: sentence by major
Descriptives
lengthofsentave11
N
natural s cience
s ocial s cience
arts and humanities
Total
19
14
15
48
Mean
14.26
7.43
10.27
11.02
Std. Deviation
15.183
8.474
10.067
12.109
Std. Error
3.483
2.265
2.599
1.748
95% Confidence Interval for
Mean
Lower Bound Upper Bound
6.94
21.58
2.54
12.32
4.69
15.84
7.50
14.54
Minimum
0
0
0
0
Maximum
60
24
36
60
ANOVA
lengthofsentave11
Between Groups
Within Groups
Total
Sum of
Squares
388.933
6502.046
6890.979
df
2
45
47
Mean Square
194.467
144.490
F
1.346
Sig.
.271
ANOVA: sentence by major
simulated data
Descriptives
lengthofsentave11
N
natural s cience
s ocial s cience
arts and humanities
Total
19
14
15
48
Mean
14.26
7.43
10.27
11.02
Std. Deviation
15.183
8.474
10.067
12.109
Std. Error
3.483
2.265
2.599
1.748
95% Confidence Interval for
Mean
Lower Bound Upper Bound
6.94
21.58
2.54
12.32
4.69
15.84
7.50
14.54
Minimum
0
0
0
0
Maximum
60
24
36
60
ANOVA
lengthofsentave11
Between Groups
Within Groups
Total
Sum of
Squares
388.933
6502.046
6890.979
df
2
45
47
Mean Square
194.467
144.490
F
1.346
Sig.
.271
ANOVA: sentence by major
simulated data
Descriptives
lengthofsentave11
N
natural s cience
s ocial s cience
arts and humanities
Total
19
14
15
48
Mean
14.26
7.43
10.27
11.02
Std. Deviation
15.183
8.474
10.067
12.109
Std. Error
3.483
2.265
2.599
1.748
95% Confidence Interval for
Mean
Lower Bound Upper Bound
6.94
21.58
2.54
12.32
4.69
15.84
7.50
14.54
Minimum
0
0
0
0
Maximum
60
24
36
60
ANOVA
lengthofsentave11
Between Groups
Within Groups
Total
Sum of
Squares
388.933
6502.046
6890.979
df
2
45
47
Mean Square
194.467
144.490
F
1.346
Sig.
.271
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).”