Statistics Project

This study applies statistical concepts on analysis of variance and repeated measures ANOVA to analyze different cases. It examines the satisfaction levels of respondents based on gender and income level, and the effect of instruction on quiz performance. The results show significant differences in satisfaction levels based on income level, but not based on gender. The number of times a test is taken also influences individual students' scores.

Statistics Project
Student Name:
Instructor Name:
Course Number:
30 May 2019

Introduction
This study sought to apply the statistical concepts on analysis of variance (one-way analysis of
variance, two-way analysis of variance and repeated measures) to analyze different cases. The
first case that the study sought to analyze was on the satisfaction levels of the respondents. The
study aimed to test whether the satisfaction levels of the respondents is affected by the gender of
the respondent or by the income level of the participant. Based on this, both one-way ANOVA
and two-way ANOVA were employed to analyze the cases. In the second part, the study aimed
to test the effect of instruction given to the students on their performance in the quizzes. Using
General Linear Model/repeated measures, a Repeated Measures ANOVA was performed to
analyze the influence of repeating the quiz over time on the quiz scores.
Results
Part 1:
a. One way ANOVA
In this section, the study sought to investigate whether there is significant difference in the
family satisfaction levels based on the income levels. The following hypothesis was tested;
Null hypothesis (H0): There is no significant difference in the average satisfaction for all the
income levels
Alternative hypothesis (HA): At least one of the income levels has different satisfaction level.
To test the above hypothesis, a one-way ANOVA was performed at 5% level of significance.
The results of the ANOVA test are presented below.
Table 1: Tests of Between-Subjects Effects
Dependent Variable: lsatisy

Source Type III
Sum of
Squares
df Mean
Square
F Sig. Partial Eta
Squared
Corrected
Model 18.619a 6 3.103 4.193 .001 .102
Intercept 1934.544 1 1934.544 2613.88
0 .000 .922
Income 18.619 6 3.103 4.193 .001 .102
Error 164.303 222 .740
Total 5492.254 229
Corrected
Total 182.923 228
a. R Squared = .102 (Adjusted R Squared = .078)
A one-way analysis of variance showed that the main effect of income was significant in
determining the satisfaction of a person, F(6, 222) = 4.19, p = .001, η2=.102. Post hoc test by
Tukey HSD showed that significant differences in satisfaction levels exist between those with
income levels of 1 and 3 (p = .002), 1 and 4 (p = .010) and 1 and 5 (p = .004).
b. Two way ANOVA table
In this section, the study sought to investigate whether there is significant difference in the
family satisfaction levels based on the income levels as well as based on gender and the
interaction effect of gender and family income level. The following three hypotheses were tested;
1. Null hypothesis (H0): There is no significant difference in the average satisfaction for all the
income levels
Alternative hypothesis (HA): At least one of the income levels has different satisfaction level.
2. Null hypothesis (H0): There is no significant difference in the average satisfaction for the
males and the females.
Alternative hypothesis (HA): There is significant difference in the average satisfaction for the
males and the females.

3. Null hypothesis (H0): There is no significant effect of the interaction between gender and
family income level on average satisfaction.
Alternative hypothesis (HA): There is significant effect of the interaction between gender and
family income level on average satisfaction.
To test the above hypotheses, a two-way ANOVA was performed at 5% level of significance.
The results of the ANOVA test are presented below.
Dependent Variable: lsatisy
Source Type III
Sum of
Squares
df Mean
Square
F Sig. Partial Eta
Squared
Corrected
Model 25.960a 13 1.997 2.735 .001 .142
Intercept 1765.279 1 1765.279 2418.00
3 .000 .918
Sex 1.999 1 1.999 2.738 .099 .013
income 18.664 6 3.111 4.261 .000 .106
sex * income 4.397 6 .733 1.004 .424 .027
Error 156.962 215 .730
Total 5492.254 229
Corrected
Total 182.923 228
a. R Squared = .142 (Adjusted R Squared = .090)
A two-way analysis of variance was performed on the influence of two independent variables
(gender and income) on the family satisfaction levels. Income level included 7 levels (0, 1, 2,
3, 4, 5 and 6) while gender consisted of two levels (male and female). The effect of gender
was found to be statistically insignificant at the 5% level of significance while the effect of
income was statistically significant at 5% level of significance. The main effect for gender
yielded an F ratio of F(1, 215) = 2.74, p = .099, η2=.013 indicating an insignificant difference
in the satisfaction levels between males and the females. For the income, the main effect

yielded an F ratio of F(6, 215) = 4.26, p = .000, η2=.013 indicating a significant difference in
the satisfaction levels for the seven different income levels. The interaction effect of gender
and income was insignificant, F(6, 215) = 1.00, p = .424, η2=.027.
Part 2
Repeated measure ANOVA
In this section, the study sought to determine whether the number of times a test is done
influence the results of individual students. The following null hypothesis was tested;
Null hypothesis (H0): the number of times a question is attended to do not affect the outcome of
individual students
Alternative hypothesis (H1): the number of times a question is attended affect the results of
individual students
To test the above, a repeated measure ANOVA was applied at a 5% level of significance and the
results displayed below.
Mauchly's Test of Sphericity indicated that the assumption of sphericity had been violated,
χ2 (9)=93.85 , p<.05, and therefore, a Greenhouse-Geisser correction was used.
Mauchly's Test of Sphericitya
Measure: results
Within
Subjects
Effect
Mauchly'
s W
Approx.
Chi-Square
df Sig. Epsilonb
Greenhouse-
Geisser
Huynh-
Feldt
Lower-
bound
instruction .400 93.851 9 .000 .640 .657 .250

From the test of within subject effect table, the Greenhouse-Geisser, the mean score for the
different times of tests were statistically different (F(18.819, 641.981)= 3.049, p
value(0.37<0.05)
This indicates that the number of times individual students performed the test influenced the final
outcome of their scores.
Tests of Within-Subjects Effects
Measure: results
Source Type III Sum
of Squares
df Mean Square F Sig.
instruction
Sphericity Assumed 18.819 4 4.705 3.049 .017
Greenhouse-Geisser 18.819 2.559 7.355 3.049 .037
Huynh-Feldt 18.819 2.629 7.159 3.049 .035
Lower-bound 18.819 1.000 18.819 3.049 .084
Error(instructio
n)
Sphericity Assumed 641.981 416 1.543
Greenhouse-Geisser 641.981 266.100 2.413
Huynh-Feldt 641.981 273.385 2.348
Lower-bound 641.981 104.000 6.173

Appendices
One-way ANOVA
Test of Homogeneity of Variances
Lsatisy
Levene
Statistic
df1 df2 Sig.
.617 6 222 .717
ANOVA
Lsatisy
Sum of
Squares
Df Mean Square F Sig.
Between
Groups 18.619 6 3.103 4.193 .001
Within Groups 164.303 222 .740
Total 182.923 228
Multiple Comparisons
Dependent Variable: lsatisy
Tukey HSD
(I)
income
(J) income Mean
Difference (I-
J)
Std.
Error
Sig. 95% Confidence Interval
Lower
Bound
Upper
Bound
0
1 .24257 .35620 .994 -.8173 1.3025
2 -.07832 .34676 1.000 -1.1101 .9535
3 -.48278 .34523 .802 -1.5100 .5445
4 -.45539 .35247 .855 -1.5042 .5934
5 -.51714 .35384 .767 -1.5700 .5357
6 -.80381 .59366 .825 -2.5703 .9627
1
0 -.24257 .35620 .994 -1.3025 .8173
2 -.32089 .18883 .617 -.8828 .2410
3 -.72535* .18602 .002 -1.2789 -.1718
4 -.69796* .19912 .010 -1.2905 -.1055
5 -.75971* .20155 .004 -1.3594 -.1600
6 -1.04638 .51754 .404 -2.5864 .4936
2 0 .07832 .34676 1.000 -.9535 1.1101
1 .32089 .18883 .617 -.2410 .8828

3 -.40446 .16724 .196 -.9021 .0932
4 -.37707 .18170 .371 -.9177 .1636
5 -.43882 .18436 .212 -.9874 .1098
6 -.72549 .51109 .791 -2.2463 .7953
3
0 .48278 .34523 .802 -.5445 1.5100
1 .72535* .18602 .002 .1718 1.2789
2 .40446 .16724 .196 -.0932 .9021
4 .02739 .17877 1.000 -.5046 .5593
5 -.03436 .18147 1.000 -.5744 .5056
6 -.32103 .51006 .996 -1.8387 1.1967
4
0 .45539 .35247 .855 -.5934 1.5042
1 .69796* .19912 .010 .1055 1.2905
2 .37707 .18170 .371 -.1636 .9177
3 -.02739 .17877 1.000 -.5593 .5046
5 -.06175 .19488 1.000 -.6416 .5181
6 -.34842 .51498 .994 -1.8808 1.1839
5
0 .51714 .35384 .767 -.5357 1.5700
1 .75971* .20155 .004 .1600 1.3594
2 .43882 .18436 .212 -.1098 .9874
3 .03436 .18147 1.000 -.5056 .5744
4 .06175 .19488 1.000 -.5181 .6416
6 -.28667 .51592 .998 -1.8218 1.2485
6
0 .80381 .59366 .825 -.9627 2.5703
1 1.04638 .51754 .404 -.4936 2.5864
2 .72549 .51109 .791 -.7953 2.2463
3 .32103 .51006 .996 -1.1967 1.8387
4 .34842 .51498 .994 -1.1839 1.8808
5 .28667 .51592 .998 -1.2485 1.8218
*. The mean difference is significant at the 0.05 level.
lsatisy
Tukey HSD
income N Subset for
alpha = 0.05
1
1 35 4.3003
0 7 4.5429
2 51 4.6212
4 40 4.9983

3 55 5.0256
5 38 5.0600
6 3 5.3467
Sig. .053
Means for groups in
homogeneous subsets are
displayed.
a. Uses Harmonic Mean Sample
Size = 11.787.
b. The group sizes are unequal.
The harmonic mean of the group
sizes is used. Type I error levels
are not guaranteed.
Two way ANOVA appendix
Between-Subjects
Factors
N
sex 1 119
2 110
income
0 7
1 35
2 51
3 55
4 40
5 38
6 3
Levene's Test of Equality of Error
Variancesa
Dependent Variable: lsatisy
F df1 df2 Sig.
.853 13 215 .604
Tests the null hypothesis that the error
variance of the dependent variable is
equal across groups.

a. Design: Intercept + sex + income +
sex * income
Tests of Between-Subjects Effects
Dependent Variable: lsatisy
Source Type III Sum
of Squares
Df Mean Square F Sig. Partial Eta
Squared
Corrected
Model 25.960a 13 1.997 2.735 .001 .142
Intercept 1765.279 1 1765.279 2418.003 .000 .918
Sex 1.999 1 1.999 2.738 .099 .013
Income 18.664 6 3.111 4.261 .000 .106
sex * income 4.397 6 .733 1.004 .424 .027
Error 156.962 215 .730
Total 5492.254 229
Corrected Total 182.923 228
a. R Squared = .142 (Adjusted R Squared = .090)
Multiple Comparisons
Dependent Variable: lsatisy
Tukey HSD
(I)
income
(J) income Mean
Difference (I-
J)
Std.
Error
Sig. 95% Confidence Interval
Lower
Bound
Upper
Bound
0
1 .2426 .35377 .993 -.8104 1.2956
2 -.0783 .34440 1.000 -1.1034 .9468
3 -.4828 .34288 .797 -1.5034 .5378
4 -.4554 .35006 .851 -1.4974 .5866
5 -.5171 .35143 .761 -1.5632 .5289
6 -.8038 .58962 .821 -2.5588 .9512
1
0 -.2426 .35377 .993 -1.2956 .8104
2 -.3209 .18755 .609 -.8791 .2373
3 -.7254* .18475 .002 -1.2753 -.1754
4 -.6980* .19776 .009 -1.2866 -.1093
5 -.7597* .20018 .004 -1.3555 -.1639
6 -1.0464 .51401 .395 -2.5763 .4836
2 0 .0783 .34440 1.000 -.9468 1.1034
1 .3209 .18755 .609 -.2373 .8791

3 -.4045 .16610 .189 -.8988 .0899
4 -.3771 .18046 .363 -.9142 .1601
5 -.4388 .18310 .205 -.9838 .1062
6 -.7255 .50761 .785 -2.2364 .7854
3
0 .4828 .34288 .797 -.5378 1.5034
1 .7254* .18475 .002 .1754 1.2753
2 .4045 .16610 .189 -.0899 .8988
4 .0274 .17755 1.000 -.5011 .5559
5 -.0344 .18024 1.000 -.5708 .5021
6 -.3210 .50658 .996 -1.8289 1.1868
4
0 .4554 .35006 .851 -.5866 1.4974
1 .6980* .19776 .009 .1093 1.2866
2 .3771 .18046 .363 -.1601 .9142
3 -.0274 .17755 1.000 -.5559 .5011
5 -.0617 .19355 1.000 -.6379 .5144
6 -.3484 .51147 .994 -1.8708 1.1740
5
0 .5171 .35143 .761 -.5289 1.5632
1 .7597* .20018 .004 .1639 1.3555
2 .4388 .18310 .205 -.1062 .9838
3 .0344 .18024 1.000 -.5021 .5708
4 .0617 .19355 1.000 -.5144 .6379
6 -.2867 .51241 .998 -1.8118 1.2385
6
0 .8038 .58962 .821 -.9512 2.5588
1 1.0464 .51401 .395 -.4836 2.5763
2 .7255 .50761 .785 -.7854 2.2364
3 .3210 .50658 .996 -1.1868 1.8289
4 .3484 .51147 .994 -1.1740 1.8708
5 .2867 .51241 .998 -1.2385 1.8118
Based on observed means.
The error term is Mean Square(Error) = .730.
*. The mean difference is significant at the .05 level.
lsatisy
Tukey HSD
Income N Subset
1
1 35 4.3003
0 7 4.5429

2 51 4.6212
4 40 4.9983
3 55 5.0256
5 38 5.0600
6 3 5.3467
Sig. .050
Means for groups in
homogeneous subsets are
displayed.
Based on observed means.
The error term is Mean
Square(Error) = .730.
a. Uses Harmonic Mean
Sample Size = 11.787.
b. The group sizes are
unequal. The harmonic mean
of the group sizes is used.
Type I error levels are not
guaranteed.
c. Alpha = .05.
Repeated measure ANOVA
Within-Subjects
Factors
Measure: results
instructio
n
Dependent
Variable
1 quiz1
2 quiz2
3 quiz3
4 quiz4
5 quiz5
Descriptive Statistics
Mean Std.
Deviation
N
quiz1 7.47 2.481 105
quiz2 7.98 1.623 105
quiz3 7.98 2.308 105

quiz4 7.80 2.280 105
quiz5 7.87 1.765 105
Multivariate Testsa
Effect Value F Hypothesis
df
Error df Sig.
instruction
Pillai's Trace .152 4.539b 4.000 101.000 .002
Wilks' Lambda .848 4.539b 4.000 101.000 .002
Hotelling's Trace .180 4.539b 4.000 101.000 .002
Roy's Largest
Root .180 4.539b 4.000 101.000 .002
a. Design: Intercept
Within Subjects Design: instruction
b. Exact statistic
Tests of Within-Subjects Effects
Measure: results
Source Type III Sum
of Squares
df Mean Square F Sig.
instruction
Sphericity Assumed 18.819 4 4.705 3.049 .017
Greenhouse-Geisser 18.819 2.559 7.355 3.049 .037
Huynh-Feldt 18.819 2.629 7.159 3.049 .035
Lower-bound 18.819 1.000 18.819 3.049 .084
Error(instructio
n)
Sphericity Assumed 641.981 416 1.543
Greenhouse-Geisser 641.981 266.100 2.413
Huynh-Feldt 641.981 273.385 2.348
Lower-bound 641.981 104.000 6.173
Mauchly's Test of Sphericitya
Measure: results
Within
Subjects Effect
Mauchly's
W
Approx. Chi-
Square
df Sig. Epsilonb
Greenhouse-
Geisser
Huynh-
Feldt
Lower-bound

instruction .400 93.851 9 .000 .640 .657 .250
Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent
variables is proportional to an identity matrix.
a. Design: Intercept
Within Subjects Design: instruction
b. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are
displayed in the Tests of Within-Subjects Effects table.
Tests of Within-Subjects Contrasts
Measure: results
Source instruction Type III Sum
of Squares
df Mean Square F Sig.
instruction
Linear 4.024 1 4.024 2.917 .091
Quadratic 8.686 1 8.686 7.858 .006
Cubic 6.095 1 6.095 2.323 .131
Order 4 .014 1 .014 .013 .910
Error(instructio
n)
Linear 143.476 104 1.380
Quadratic 114.956 104 1.105
Cubic 272.905 104 2.624
Order 4 110.644 104 1.064
Tests of Between-Subjects Effects
Measure: results
Transformed Variable: Average
Source Type III Sum
of Squares
df Mean
Square
F Sig.
Intercept 32097.190 1 32097.190 1974.033 .000
Error 1691.010 104 16.260