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04/03/2021 1
One Sample and Paired Sample t-Tests, and Group Comparison
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Table of Contents
A7.1: Chapter 10, Problem 10.1, One‐Sample t-Test ...................................................................3
A7.2: Chapter 10, Problem 10.2, Independent Samples t‐Test .....................................................4
A7.3: Chapter 10, Problem 10.3, the Nonparametric Mann‐Whitney U Test ................................6
A7.4: Chapter 10, Problem 10.4, Paired Samples t-Test...............................................................7
A7.5: Chapter 10, Problem 10.5, Nonparametric Wilcoxon Test for Two Related Samples .........9
A7.6, Application Problem ‐ Comparing Two Groups ............................................................... 10
Exploring the Difference between Genders on Student’s Average Height .............................. 10
Exploring the Differences between Raters in the mosaic pattern test score between rater 1 and
rater 2 .................................................................................................................................... 11
References ................................................................................................................................ 14
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A7.1: Chapter 10, Problem 10.1, One‐Sample t-Test
One-sample T-test is used to indicate whether there exist any differences between the
sample mean and an arbitrary set value for comparisons reasons. The assumptions made before
conducting this test are that the dependent variable is normally distributed and that the predictor
variables are independent of each other (Morgan et al.,2013). The One sample t-Test is conducted
to investigate whether there exist any significant differences between the scholastic aptitude Test-
Math, and the test value 500. The following are the corresponding results done using SPSS analysis
part;
One-Sample Statistics
N
Mean
Std.
Deviation
Std. Error
Mean
scholastic aptitude test -
math
75
490.53
94.553
10.918
One-Sample Test
Test Value = 500
t
df
Sig. (2-
tailed)
Mean
Difference
95% Confidence Interval of
the Difference
Lower
Upper
scholastic aptitude test
- math
-.867
74
.389
-9.467
-31.22
12.29
The mean of the scholastic aptitude test- Math is 490.53 with a standard deviation of
94.553. The analysis results indicate that t=-0.867, p=0.389, which indicates that there exist no
significant differences between the Scholastic aptitude test scores and the set value 500. The 95%
confidence interval gives the threshold in which the mean lies, which is -32 and 12.29, on the
lower and upper side, respectively.
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04/03/2021 4
A7.2: Chapter 10, Problem 10.2, Independent Samples t‐Test
The independence samples t-tests also tests the significant differences in means given that
the set assumptions of independence are met. The main assumptions made while testing
differences in unrelated groups is that the variances of the two groups are similar (Morgan et al.,
2013). The other assumption is that the depend variables in both populations under comparison is
normally distributed, and the scores of the two groups are all independent (Morgan et al., 2013).
The sample analysis done here is to determine whether there exist significant differences in the
math achievement, grades in high school, and visualization test in respect to gender of the student.
Group Statistics
gender
N
Mean
Std. Deviation
Std. Error Mean
grades in High school
male
34
5.50
1.638
.281
female
41
5.83
1.515
.237
math achievement test
male
34
14.7550
6.03154
1.03440
female
41
10.7479
6.69612
1.04576
visualization test
male
34
6.4265
4.47067
.76671
female
41
4.2622
3.10592
.48506
The grades in high school, math achievement tests score and visualization test scores are
shown with male and female categorization specified. The means for the respective variables
between male and females are therefore under comparisons. The Tables on independent samples
Tests indicates the differences in association between the means between the aforementioned
groups. The tests between gender of the students and grades in high school has the result t=-0.903,
df=73, p= 0.369, >0.05. This implies that there are no significant differences in the means of the
high school grades among female and male students.
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Independent Samples Test
Levene's Test
for Equality
of Variances
t-test for Equality of Means
F
Sig.
t
df
Sig. (2-
tailed)
Mean
Differe
nce
Std.
Error
Differe
nce
95% Confidence
Interval of the
Difference
Lower
Upper
grades in h.s.
Equal
variances
assumed
.574
.451
-.903
73
.369
-.329
.365
-1.056
.397
Equal
variances
not
assumed
-.897
68.145
.373
-.329
.367
-1.062
.403
math
achievement test
Equal
variances
assumed
.537
.466
2.697
73
.009
4.00704
1.4854
8
1.04648
6.96760
Equal
variances
not
assumed
2.724
72.472
.008
4.00704
1.4709
2
1.07515
6.93894
visualization test
Equal
variances
assumed
6.510
.013
2.466
73
.016
2.16428
.87778
.41486
3.91369
Equal
variances
not
assumed
2.385
57.150
.020
2.16428
.90727
.34761
3.98094
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The tests between gender of the students and math achievement test scores has the result t=2.697,
df=73, p= 0.009, <0.05. This implies that there exist significant differences in the math
achievement scores among female and male students. The tests between gender of the students and
visualization test scores has the result t=2.466, df =73, p= 0.016, <0.05. This means that there exist
significant differences in the visualization test scores among female and male students.
A7.3: Chapter 10, Problem 10.3, the Nonparametric Mann‐Whitney U Test
The non-parametric tests are essential especially for variables that do not follow the normal
distribution. The Mann-Whitney U Test is used with a between-groups design with two levels of
the independent variable (Morgan et al., 2013). The main assumptions for this test is that the
dependent variable is continuous and that the variables under comparisons are independent of each
other. The following are the non-parametric tests results between gender, maths achievement tests
scores, grades in high school, and visualization tests scores;
Ranks
gender
N
Mean Rank
Sum of Ranks
grades in h.s.
male
34
35.78
1216.50
female
41
39.84
1633.50
Total
75
math achievement test
male
34
45.10
1533.50
female
41
32.11
1316.50
Total
75
visualization test
male
34
43.65
1484.00
female
41
33.32
1366.00
Total
75
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Test Statisticsa
grades in h.s.
math achievement test
visualization test
Mann-Whitney U
621.500
455.500
505.000
Wilcoxon W
1216.500
1316.500
1366.000
Z
-.818
-2.575
-2.052
Asymp. Sig. (2-tailed)
.413
.010
.040
a. Grouping Variable: gender
The tests indicate similar results as the ones done using the parametric tests. The grouping
variable gender has a diversified significantly differences in means for the math achievement test
scores and visualization tests. For the case of grades in high school, the mean of the 41 females
(39.84) was highly ranked with U=621.500, p=0.413, hence indicating insignificant results which
means there are no differences in the grades between male and female students.
A7.4: Chapter 10, Problem 10.4, Paired Samples t-Test
The paired t-Tests are also measures of comparisons between two variables on the same
level. The assumptions considered in this case is that the dependent variable is normally distributed
and also dichotomous (Morgan et al., 2013). The following are the corresponding results
Paired Samples Statistics
Mean
N
Std.
Deviation
Std. Error
Mean
Pair 1
father's education
4.73
73
2.830
.331
mother's
education
4.14
73
2.263
.265
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Paired Samples Correlations
N
Correlation
Sig.
Pair 1
father's education & mother's
education
73
.681
.000
Paired Samples Test
Paired Differences
t
df
Sig. (2-
tailed)
Mean
Std.
Deviation
Std. Error
Mean
95% Confidence
Interval of the
Difference
Lower
Upper
Pair
1
father's education
- mother's
education
.589
2.101
.246
.099
1.079
2.396
72
.019
The correlation coefficient between father’s education and mother’s education is 0.681
which indicates a positive high significant association between the pair. The paired samples tests
shows that the p-value is 0.019 which indicates that mothers and father’ education are all
significantly different.
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A7.5: Chapter 10, Problem 10.5, Nonparametric Wilcoxon Test for Two Related Samples
The paired samples test is done considering the non-parametric test values to confirm the
existence of significant differences between groups. The following are the results for the analysis;
Ranks
N
Mean Rank
Sum of Ranks
mother's education - father's
education
Negative Ranks
27a
29.20
788.50
Positive Ranks
21b
18.45
387.50
Ties
25c
Total
73
visualization 2 - visualization
test
Negative Ranks
24d
37.25
894.00
Positive Ranks
37e
26.95
997.00
Ties
14f
Total
75
a. mother's education < father's education
b. mother's education > father's education
c. mother's education = father's education
d. visualization 2 < visualization test
e. visualization 2 > visualization test
f. visualization 2 = visualization test
Test Statisticsc
mother's education - father's
education
visualization 2 - visualization
test
Z
-2.085a
-.373b
Asymp. Sig. (2-tailed)
.037
.709
a. Based on positive ranks.
b. Based on negative ranks.
c. Wilcoxon Signed Ranks Test
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The analysis results show that there are no significant differences in means for the visualization
tests, while there are significant differences between the mothers’s and father’s education.
A7.6, Application Problem ‐ Comparing Two Groups
Exploring the Difference between Genders on Student’s Average Height
The independent samples t-test is used to test the differences in height by gender for college
students. The following is the research question and hypothesis;
Research Question; Are there any differences in the average students’ height based on the specific
gender orientation?
Null hypothesis; There exist no significant differences in the average students’ height based on
the specific gender orientation.
Alternative hypothesis; There exists significant differences in the average students’ height based
on the specific gender orientation.
The analysis is appropriate because all the assumptions have been met; the dependent
variable is normally distributed, while the variables under comparisons (Predictors), are all
independent of each other. The following are the analysis results;
Group Statistics
MaleGen
N
Mean
Std. Deviation
Std. Error Mean
student height in inches
Not Male
25
64.4400
2.56710
.51342
male
25
70.1600
2.83843
.56769
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04/03/2021 11
Independent Samples Test
Levene's Test for
Equality of
Variances
t-test for Equality of Means
F
Sig.
t
df
Sig. (2-
tailed)
Mean
Differenc
e
Std. Error
Differenc
e
95%
Confidence
Interval of the
Difference
Lower
Uppe
r
student height in
inches
Equal
variance
s
assumed
.154
.697
-7.473
48
.000
-5.72000
.76542
-7.25898
-
4.181
02E0
Equal
variance
s not
assumed
-7.473
47.523
.000
-5.72000
.76542
-7.25938
-
4.180
62E0
The analysis results show that the p-value corresponding to the students’ height in inches in
comparison to gender is 0.00. Therefore, the null hypothesis is rejected and it is concluded that
there exists significant differences in the average students’ height based on the specific gender
orientation.
Exploring the Differences between Raters in the mosaic pattern test score between rater 1
and rater 2
The following are the research questions and hypothesis for this test;
Research questions; Are there any significant differences or association between the
mosaic pattern score and the raters (motivation and pleasure scales)?
Null Hypothesis: There exists no significant differences or association between the mosaic
pattern score and the raters (motivation and pleasure scales).
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Alternative Hypothesis: There exists significant differences or association between the
mosaic pattern score and the raters (motivation and pleasure scales).
The following are the results for these tests;
Paired Samples Statistics
Mean
N
Std.
Deviation
Std. Error
Mean
Pair 1
mosaic, pattern
test
27.399
74
9.6383
1.1204
item01 motivation
2.96
74
.928
.108
Pair 2
mosaic, pattern
test
27.413
75
9.5738
1.1055
item02 pleasure
3.52
75
.906
.105
Paired Samples Correlations
N
Correlation
Sig.
Pair 1
mosaic, pattern test & item01
motivation
74
.149
.206
Pair 2
mosaic, pattern test & item02
pleasure
75
.046
.697
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04/03/2021 13
Paired Samples Test
Paired Differences
t
df
Sig. (2-
tailed)
Mean
Std.
Deviation
Std. Error
Mean
95% Confidence
Interval of the
Difference
Lower
Upper
Pair
1
mosaic, pattern
test - item01
motivation
24.439
2
9.5444
1.1095
22.2279
26.6504
22.027
73
.000
Pair
2
mosaic, pattern
test - item02
pleasure
23.893
3
9.5752
1.1056
21.6903
26.0964
21.610
74
.000
The results indicate positive but weak correlation between the mosaic pattern tests and the
two rates; motivation and pleasure scales. The paired samples t-test has p-values equal to 0.00
which are all less than 0.05 for both pairs. Therefore, the null hypothesis is rejected an it is
concluded that there exists significant differences or association between the mosaic pattern score
and the raters (motivation and pleasure scales).
Course Name & Assignment Number
04/03/2021 14
References
Morgan, G., Leech, N., Gloeckner, G., Barrett, K. (2013). IBM SPSS for Introductory Statistics
(5th Ed.). New York, NY
