Chapter 12
Quantitative Data Analysis: Hypothesis Testing
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Type I Errors, Type II Errors and Statistical
Power
 Type I error (): the probability of rejecting the null
hypothesis when it is actually true.
 Type II error (): the probability of failing to reject the
null hypothesis given that the alternative hypothesis is
actually true.
 Statistical power (1 - ): the probability of correctly
rejecting the null hypothesis.
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Choosing the Appropriate Statistical Technique
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Testing Hypotheses on a Single Mean
 One sample t-test: statistical technique that is used to
test the hypothesis that the mean of the population from
which a sample is drawn is equal to a comparison
standard.
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Testing Hypotheses about Two Related
Means
 Paired samples t-test: examines differences in same
group before and after a treatment.
 The Wilcoxon signed-rank test: a non-parametric test
for examining significant differences between two
related samples or repeated measurements on a single
sample. Used as an alternative for a paired samples ttest when the population cannot be assumed to be
normally distributed.
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Testing Hypotheses about Two Related
Means - 2
 McNemar's test: non-parametric method used on
nominal data. It assesses the significance of the
difference between two dependent samples when the
variable of interest is dichotomous. It is used primarily in
before-after studies to test for an experimental effect.
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Testing Hypotheses about Two Unrelated
Means
 Independent samples t-test: is done to see if there are
any significant differences in the means for two groups
in the variable of interest.
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Testing Hypotheses about Several Means
 ANalysis Of VAriance (ANOVA) helps to examine the
significant mean differences among more than two
groups on an interval or ratio-scaled dependent
variable.
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Regression Analysis
 Simple regression analysis is used in a situation where
one metric independent variable is hypothesized to
affect one metric dependent variable.
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Scatter plot
100
LKLHD_DATE
80
60
40
20
30
40
50
60
70
80
90
PHYS_ATTR
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Simple Linear Regression
Yi   0  1  X i   i
Y
ˆ0
ˆ1
? `0
1
ˆ0
X
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Ordinary Least Squares Estimation
n
Minimize
e
2
i
i 1
Yi
Yˆi
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ei
Xi
12
SPSS
Analyze  Regression  Linear
Model Summary
Model
1
R
.841
R Square
.707
Adjusted
R Square
.704
Std. Error of
the Estimate
5.919
ANOVA
Model
1
Regression
Residual
Total
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Sum of
Squares
8195.319
3398.640
11593.960
df
1
97
98
Mean Square
8195.319
35.038
F
233.901
Sig.
.000
13
SPSS cont’d
Coefficients
Model
1
(Constant)
PHYS_ATTR
Unstandardized
Coefficients
B
Std. Error
34.738
2.065
.520
.034
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Standardized
Coefficients
Beta
.841
t
16.822
15.294
Sig.
.000
.000
14
Model validation
1.
2.
Face validity: signs and magnitudes make sense
Statistical validity:
–
–
–
–
–
3.
Model fit: R2
Model significance: F-test
Parameter significance: t-test
Strength of effects: beta-coefficients
Discussion of multicollinearity: correlation matrix
Predictive validity: how well the model predicts
–
Out-of-sample forecast errors
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SPSS
Model Summary
Model
1
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R
.841
R Square
.707
Adjusted
R Square
.704
Std. Error of
the Estimate
5.919
16
Measure of Overall Fit: R2
 R2 measures the proportion of the variation in y that is explained by the
variation in x.
 R2 = total variation – unexplained variation
total variation
 R2 takes on any value between zero and one:
– R2 = 1: Perfect match between the line and the data points.
– R2 = 0: There is no linear relationship between x and y.
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SPSS
Model Summary
Model
1
R
.841
R Square
.707
Adjusted
R Square
.704
Std. Error of
the Estimate
5.919
= r(Likelihood to Date, Physical Attractiveness)
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Model Significance
 H0: 0 = 1 = ... = m = 0
(all parameters are zero)
H1: Not H0
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Model Significance
 H0: 0 = 1 = ... = m = 0
H1: Not H0
(all parameters are zero)
 Test statistic (k = # of variables excl. intercept)
F =
(SSReg/k)
(SSe/(n – 1 – k)
~ Fk, n-1-k
SSReg = explained variation by regression
SSe = unexplained variation by regression
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SPSS
ANOVA
Model
1
Regression
Residual
Total
Sum of
Squares
8195.319
3398.640
11593.960
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df
1
97
98
Mean Square
8195.319
35.038
F
233.901
Sig.
.000
21
Parameter significance
 Testing that a specific parameter is significant
(i.e., j  0)
 H0 :  j = 0
H1 :  j  0
 Test-statistic: t = bj/SEj ~ tn-k-1
with bj = the estimated coefficient for j
SEj = the standard error of bj
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SPSS cont’d
Coefficients
Model
1
(Constant)
PHYS_ATTR
Unstandardized
Coefficients
B
Std. Error
34.738
2.065
.520
.034
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Standardized
Coefficients
Beta
.841
t
16.822
15.294
Sig.
.000
.000
23
Conceptual Model
Physical
Attractiveness
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+
Likelihood
to Date
24
Multiple Regression Analysis
 We use more than one (metric or non-metric)
independent variable to explain variance in a (metric)
dependent variable.
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Conceptual Model
Perceived
Intelligence
Physical
Attractiveness
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+
+
Likelihood
to Date
26
Model Summary
Model
1
R
.844
R Square
.712
Adjusted
R Square
.706
Std. Error of
the Estimate
5.895
ANOVA
Model
1
Regression
Residual
Total
Sum of
Squares
8257.731
3336.228
11593.960
df
2
96
98
Mean Square
4128.866
34.752
F
118.808
Sig.
.000
Coefficients
Model
1
(Constant)
PERC_INTGCE
PHYS_ATTR
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Unstandardized
Coefficients
B
Std. Error
31.575
3.130
.050
.037
.523
.034
Standardized
Coefficients
Beta
.074
.846
t
10.088
1.340
15.413
Sig.
.000
.183
.000
27
Conceptual Model
Gender
Perceived
Intelligence
Physical
Attractiveness
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+
+
+
Likelihood
to Date
28
Moderators
 Moderator is qualitative (e.g., gender, race, class) or quantitative (e.g., level
of reward) that affects the direction and/or strength of the relation between
dependent and independent variable
 Analytical representation
Y = ß0 + ß1X1 + ß2X2 + ß3X1X2
with
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Y = DV
X1 = IV
X2 = Moderator
29
Moderators
Model Summary
Model
1
R
.910
R Square
.828
Adjusted
R Square
.821
Std. Error of
the Estimate
4.601
ANOVA
Model
1
Regression
Residual
Total
Sum of
Squares
9603.938
1990.022
11593.960
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df
4
94
98
Mean Square
2400.984
21.170
F
113.412
Sig.
.000
30
Moderators
Coefficients
Model
1
(Constant)
PERC_INTGCE
PHYS_ATTR
GENDER
PI_GENDER
Unstandardized
Coefficients
B
Std. Error
32.603
3.163
.000
.043
.496
.027
-.420
3.624
.127
.058
Standardized
Coefficients
Beta
.000
.802
-.019
.369
t
10.306
.004
18.540
-.116
2.177
Sig.
.000
.997
.000
.908
.032
interaction significant effect on dep. var.
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Conceptual Model
Gender
Perceived
Intelligence
+
+
+
Physical
Attractiveness
Likelihood
to Date
+
Communality of
Interests
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+
Perceived Fit
32
Mediating/intervening variable

Accounts for the relation between the independent and dependent variable

Analytical representation
1.
Y = ß0 + ß1 X
=> ß1 is significant
2.
M = ß2 + ß3 X
=> ß3 is significant
3.
Y = ß4 + ß5X + ß6M
=> ß5 is not significant
=> ß6 is significant
With
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Y = DV
X = IV
M = mediator
33
Step 1
Mode l Summary
Model
1
R
.963
R Square
.927
Adjus ted
R Square
.923
Std. Error of
the Estimate
3.020
ANOVA
Model
1
Regression
Residual
Total
Sum of
Squares
10745.603
848.357
11593.960
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df
5
93
98
Mean Square
2149.121
9.122
F
235.595
Sig.
.000
34
Step 1 cont’d
Coefficients
Model
1
(Cons tant)
PERC_INTGCE
PHYS_ATTR
GENDER
PI_GENDER
COMM_INTER
Unstandardized
Coefficients
B
Std. Error
17.094
2.497
.030
.029
.517
.018
-.783
2.379
.122
.038
.212
.019
Standardized
Coefficients
Beta
.044
.836
-.036
.356
.319
t
6.846
1.039
29.269
-.329
3.201
11.187
Sig.
.000
.301
.000
.743
.002
.000
significant effect on dep. var.
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Step 2
Mode l Summary
Model
1
R
.977
Adjus ted
R Square
.955
R Square
.955
Std. Error of
the Estimate
2.927
ANOVA
Model
1
Regression
Residual
Total
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Sum of
Squares
17720.881
831.079
18551.960
df
1
97
98
Mean Square
17720.881
8.568
F
2068.307
Sig.
.000
36
Step 2 cont’d
Coefficients
Model
1
(Cons tant)
COMM_INTER
Unstandardized
Coefficients
B
Std. Error
8.474
1.132
.820
.018
Standardized
Coefficients
Beta
.977
t
7.484
45.479
Sig.
.000
.000
significant effect on mediator
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Step 3
Mode l Summary
Model
1
R
.966
R Square
.934
Adjus ted
R Square
.930
Std. Error of
the Estimate
2.885
ANOVA
Model
1
Regression
Residual
Total
Sum of
Squares
10828.336
765.624
11593.960
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df
6
92
98
Mean Square
1804.723
8.322
F
216.862
Sig.
.000
38
Step 3 cont’d
Coefficients
Model
1
(Cons tant)
PERC_INTGCE
PHYS_ATTR
GENDER
PI_GENDER
COMM_INTER
PERC_FIT
Unstandardized
Coefficients
B
Std. Error
14.969
2.478
.019
.028
.518
.017
-2.040
2.307
.142
.037
-.051
.085
.320
.102
Standardized
Coefficients
Beta
.028
.839
-.094
.412
-.077
.405
t
6.041
.688
30.733
-.884
3.825
-.596
3.153
Sig.
.000
.493
.000
.379
.000
.553
.002
insignificant effect of indep. var on dep. Var.
significant effect of mediator on dep. var.
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39