Quantitative Methods & Individual Differences

This study focuses on statistical exercises and analysis of four study problems. Each task was appropriately analysed and the results are presented in APA style. Test of assumptions was carried out before each statistical analysis, and relevant statistical information together with appropriate estimates has been reported. Appropriate set of hypotheses with proper choice of statistical test has been clearly stated.

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Quantitative Methods & Individual Differences

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Table of Contents
Introduction................................................................................................................................4
Statistical Exercises....................................................................................................................4
Exercise 1...................................................................................................................................4
Hypotheses.............................................................................................................................4
Results....................................................................................................................................5
Variables.............................................................................................................................5
Choice of test......................................................................................................................5
Testing of Assumptions......................................................................................................5
Findings..............................................................................................................................5
Discussion...............................................................................................................................6
Appendices - 1........................................................................................................................6
Country wise analysis.........................................................................................................6
Sex wise analysis................................................................................................................8
Exercise 2.................................................................................................................................10
Hypotheses...........................................................................................................................10
Results..................................................................................................................................11
Variables...........................................................................................................................11
Choice of test....................................................................................................................11
Testing of Assumptions....................................................................................................11
Findings............................................................................................................................12
Discussion.............................................................................................................................13
Appendices – 2.....................................................................................................................13
Exercise 3.................................................................................................................................16
Hypotheses...........................................................................................................................16
Results..................................................................................................................................17
Variables...........................................................................................................................17

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Choice of test....................................................................................................................17
Testing of Assumptions....................................................................................................17
Findings............................................................................................................................18
Discussion.............................................................................................................................19
Appendices - 3......................................................................................................................19
Exercise 4.................................................................................................................................22
Hypotheses...........................................................................................................................22
Results..................................................................................................................................22
Variables...........................................................................................................................22
Choice of test....................................................................................................................22
Testing of Assumptions....................................................................................................22
Findings............................................................................................................................23
Discussion.............................................................................................................................24
Appendices - 4......................................................................................................................24
References................................................................................................................................26

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Introduction
In this study of statistical exercises, statistical analysis of four study problems was
carried out. Each task was appropriately analysed and the results are presented in APA style
in line the module. Test of assumptions was carried out before each statistical analysis, and
relevant statistical information together with appropriate estimates has been reported.
Appropriate set of hypotheses with proper choice of statistical test has been clearly stated.
Each exercise has been answered with checking of assumptions, construction of hypotheses,
results analysis, and discussion with respect to existing literatures.
Statistical Exercises
Exercise 1
Hypotheses
Countries:
Null hypothesis: H01 : M A =M B : There is no difference in median of “life
satisfaction” between people of Argentina and Brazil.
Alternate hypothesis: H A 1 : M A≠ M B : There is significant difference in median of
“life satisfaction” between people of Argentina and Brazil (two tailed).
Sexes:
Null hypothesis: H02 : M m=M f : There is no difference in median of “life
satisfaction” between male and females.
Alternate hypothesis: H A 2 : M m≠M f : There is significant difference in median of
“life satisfaction” between male and females (two tailed).

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Results
Variables
DV: “life satisfaction” (Ordinal), IVs: Country and Sex (both categorical)
Choice of test
Due to non-normality of the distribution of “life satisfaction” and presence of outliers,
Kruskal-Wallis Test (non-parametric) was chosen, although, homogeneity of variances for
both the categorical variables was satisfied.
Testing of Assumptions
Shapiro Wilk test confirmed that distribution of “life satisfaction” was not normal for
Argentina (W (199) = 0.93, p < 0.01) and Brazil (W (98) = 0.88, p < 0.01). Presence of six
outlier observations was detected for Argentina. Levene’s test for equality of variances
implied that there was homogeneity of variances (F (199) = 0.52, p = 0.47) between “life
satisfaction” of two countries.
Shapiro Wilk test confirmed that distribution of “life satisfaction” was not normal for
men (W (150) = 0.92, p < 0.01) and women (W (147) = 0.92, p < 0.01). Presence of two
outlier observations was detected for males. Levene’s test for equality of variances implied
that there was homogeneity of variances (F (98) = 0.40, p = 0.53) between “life satisfaction”
of two sexes.
Findings
Median “life satisfaction” in Argentina and Brazil, as well for both the sexes has been
provided in Table 1.
Table 1: Median and Interquartile Range of “life satisfaction” for Independent Variables (Countries and
Sexes)
Countries Sex
Argentina Brazil Male Female
7.00 (3.00) 8.00 (4.00) 8.00 (3.00) 8.00 (3.00)

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“Life satisfaction” was significantly different by countries H ( 1) =6 . 93, p < 0 .01 .
Pairwise comparisons revealed that there was a significant difference in “life satisfaction”
between the people of Argentina and Brazil (U = 7944, p = .008).
Also, there was no significant difference in “life satisfaction” by genders
H (1) =0 . 12, p = 0 . 73 . Pairwise comparisons also revealed that there was no significant
difference in “life satisfaction” between male and females (U = 10773.5, p = 0.73).
Discussion
According to results of the present study, Median “life satisfaction” was different
across countries, but not across genders. In previous literatures, life satisfaction also has been
experienced in accordance to quality of life, religion, and social structure. Senik (2011) using
novel survey in 21 European countries addressed that cognitive and hedonic measures of
well-being reflect individual’s quality of life. In another study, Okulicz-Kozaryn (2010)
discovered that religious people are considerably happier in life, which also depends on social
structure of a country. Elgar et al. (2011) researched on a four-factor measure of social capital
revealing in a multilevel analyses that “life satisfaction” is dependent on country’s social
capital and health.
The result of this study was also in contradiction with some previous literatures.
However, Elgar et al. (2011), indicated that benefits of social capital matters less to men
compared to that of women, indicating a higher “life satisfaction” in women with better
social benefits. Ye, Yu, & Li (2012) showed that males scored higher compared to females on
“life satisfaction”. No such result was supported by the present work.
Appendices - 1
Country wise analysis
Descriptive Statistics

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Test for Normality
Side-by-side Box plot

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Test for Homogeneity
Kruskal Wallis and Mann Whitney U test results
Sex wise analysis
Descriptive Statistics

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Test for Normality
Side-by-side Box plot

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Test for Homogeneity
Kruskal Wallis and Mann Whitney U test results
Exercise 2
Hypotheses
Person’s Sex
Null hypothesis: H01 : β1=0 : There is no linear relation between a person’s sex and
his/her “life satisfaction”.
Alternate hypothesis: H A 1 : β1≠0 : There is a significantly linear relation between a
person’s sex and his/her “life satisfaction”.

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Freedom of choice
Null hypothesis: H02 : β2=0 : There is no linear relation between a person’s freedom
of choice and his/her “life satisfaction”.
Alternate hypothesis: H A 2 : β2≠0 : There is a significantly linear relation between a
person’s freedom of choice and his/her “life satisfaction”.
Results
Variables
DV: “life satisfaction” (Ordinal), IVs: Sex (nominal) and freedom of choice (ordinal)
Choice of test
A multiple linear regression model with sex and freedom of choice as predictors and
“life satisfaction” as outcome variable is the proper choice of test, since there were two
predictors and one outcome variable.
Testing of Assumptions
A significant linear correlation between freedom of choice and “life satisfaction” was
noted ( r=0. 49 , p<0 . 01 ) . Pearson’s correlation was not an appropriate choice of test for
measuring association between a nominal and an ordinal variable. Hence, Pearson’s
correlation was not used to find linear relation between person’s sex and “life satisfaction”.
Kendal’s tau_b between sex and “life satisfaction” indicated a non-significant correlation
( taub=−0 . 04 , p=0. 34 ) .
Unstandardized residuals were found to violate assumption of normal distribution
( W (395 )=0 . 98 , p< 0 .01 ) , due to presence of few outliers.
Correlation between “life satisfaction” and freedom of choice
( taub=−0 . 004 , p=0. 93 ) and the variance inflation factor for both the predictors was 1.
Hence, no multicollinearity was noted.
Scatter plot of standardized residuals and predicted values in Figure 1 indicated that
variance of errors were equally and randomly distributed without following any pattern.
Hence, no heteroscedasticity was noted.

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Figure 1: Scatter plot of standardized residuals and predicted values
Findings
Average “life satisfaction” was 6.85 (SD = 2.27), and average freedom of choice was
6.77 (SD = 2.31). The linear regression model was found to be statistically significant
( F ( 2 , 392 ) =63 . 80 , p< 0. 01 ) . Coefficient of determination ( R2=0 .25 , RAdj
2 =0 .24 ) indicated
that both the predictors were able to explain almost 24% variation of “life satisfaction”.
A person’s sex was noted to be an insignificant linear predictor of “life satisfaction”
( β1=−0 .20 , t=−1 . 03 , p=0 .30 ) . Hence, the null hypothesis H01 failed to get rejected at 5%
level of significance.
Freedom of choice was a statistically significant linear predictor
( β2=0 . 48 , t=11. 24 , p<0 . 01 ) and the null hypothesis H02 was rejected. Hence, freedom of
choice was a significant predictor of “life satisfaction”. Therefore, average change of freedom
of choice by one unit is expected to modify average “life satisfaction” by 0.48 grades.
The final regression model was evaluated to be: Life Satisfaction = 3.88 -0.20*Sex +
0.48*Freedom of Choice.

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Discussion
“Life satisfaction” was regressed over gender and freedom of choice. Gender was
unable to explain variation in “life satisfaction”, but, freedom of choice had a significant and
linear association with “life satisfaction” of a person. Previously Freedom of choice in terms
of liberty to internet usage was found to be a significant predictor of life satisfaction. Pénard,
Poussing, and Suire (2013) found evidence of internet users being highly satisfied in their life
compared to non-internet users, in the European Value Survey. Again, regarding impact of
liberation, Welzel and Inglehart (2010) gave stronger emphasis on liberation values and
found that intervention greatly shapes people’s “life satisfaction”. In another study related to
effect of freedom or autonomy on “life satisfaction”, Delhey (2010) investigated the “post-
material” concerns for happiness, and found that personal autonomy serve as an indicator for
“life satisfaction”. Also, Inglehart (2010) researched on the differences in the social
perspective of Well-Being satisfaction, and stated that people incline toward the happiness in
life by making best use of free choice. The present research outcome was in line with these
literature outcomes.
Appendices – 2
Descriptive Statistics with Pearson’s Correlation

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Regression Model with ANOVA

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Residual Histogram
Residual Descriptive and Normality

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Predicted versus Residuals
Box plot of residuals
Exercise 3
Hypotheses
A. Null hypothesis: ( H01 : μL=μW ) : Average importance of leisure and average importance
of work for a person’s life were equal.
Alternate hypothesis: ( H A 1 : μL≠μW ) : Average importance of leisure and average
importance of work for a person’s life were significantly different.

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B. Null hypothesis: H02 : Sex of a person does not have any main impact on importance of
leisure in life.
Alternate hypothesis: H A 2 : Sex of a person does have a significant impact on
importance of leisure in life.
C. Null hypothesis: H03 : Sex of a person does not have any main impact on importance of
work in life.
Alternate hypothesis: H A 3 : Sex of a person does have a significant impact on
importance of work in life.
Results
Variables
A. DVs: Importance of leisure and importance of work in life (ordinal variables)
B. DV: Importance of leisure in life, IV: Sex (nominal)
C. DV: Importance of work in life, IV: Sex (nominal)
Choice of test
A. One-way dependent ANOVA (As Importance of leisure and importance of work in
life were both dependent variables and their normality has been discussed in testing of
assumptions).
B. Multivariate ANOVA (As impact of nominal variable (sex) on continuous outcome
variable (importance of leisure) was checked).
C. Multivariate ANOVA (As check impact of nominal variable (sex) on continuous
outcome variable (importance of work) was checked).
Testing of Assumptions
A. Distribution of importance of leisure ( W =0. 96 , p<0 . 01 ) and importance of work
( W =0. 95 , p<0 . 01 ) in life were not normal, and were slightly positive skewed. The
sample size in both cases was large enough (N = 395 > 30) for using Central Limit
theorem to assume normality. No outliers were present for both the DVs. Due to two
dependent variables Mauchly's test of Sphericity was not useful. Assumption of
independence of the observations was understandable in nature.

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B. Importance of leisure in life was ordinal, and due to sample size was considered to be
interval type data. Gender wise distribution of importance of leisure for men
( W =0. 96 , p<0 . 01 ) and women ( W =0. 96 , p<0 . 01 ) in life were not normal, and were
slightly positive skewed. The sample size in both cases was large enough (N > 30) for
using Central Limit theorem to assume normality. No outlier for DV was present for
both the levels of IV. Levene’s test based on means revealed
( F ( 1 , 393 ) =0 . 74 , p=0 . 39 ) that variances for both sexes were statistically equal
(assumption of homogeneity).
C. Importance of work in life was ordinal, and due to sample size was considered to be
interval type data. Gender wise distribution of importance of leisure for men
( W =0. 95 , p<0 . 01 ) and women ( W =0. 93 , p<0 . 01 ) in life were not normal, and were
slightly positive skewed. The sample size in both cases was large enough (N >30) for
using Central Limit theorem to assume normality. No outlier for DV was present for
both the levels of IV. Levene’s test based on means revealed
( F ( 1 , 393 ) =0 . 05 , p=0 . 83 ) that variances for both sexes were statistically equal
(assumption of homogeneity).
Findings
Average of importance of leisure and work in life has been presented in Table 2.
Table 2: Average and Standard Deviation of importance of leisure and work in life
DV N M SD
Leisure 395 6.10 1.95
Work 395 3.61 1.50
Leisure _Male 193 6.15 1.90
Leisure _Female 202 6.06 2.00
Work _Male 193 3.73 1.52
Work _Female 202 3.49 1.47
A. Importance of leisure in life ( M =6 . 10 , D=1. 95 ) was found to be statistically more
important than importance of work in life ( M =3 . 61, SD=1 .50 ) from One-way

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dependent ANOVA ( F ( 1 , 394 ) =1074 , p <0 . 01, η2=0 . 73 ) . Therefore, the null
hypothesis H01 was rejected at 1% level of significance.
B. Sex of a person was found to have no significant main impact on importance of
leisure in life in a Multivariate ANOVA ( F ( 1 , 393 ) =0. 21 , p=0 . 64 , η2=0 .001 ) .
Therefore, the null hypothesis H02 failed to get rejected at 5% level of significance.
C. Sex of a person was found to have no significant main impact on importance of work
in life in a Multivariate ANOVA ( F ( 1 , 393 ) =2 . 45 , p=0 . 12 ,η2=0 . 006 ) . Therefore,
the null hypothesis H03 failed to get rejected at 5% level of significance.
Therefore, there was no sex difference between importance to leisure or work.
Discussion
People were found to emphasis more on importance of leisure in life compared to
importance of work in their life. The result was consistent across both the genders, and no
statistical impact of sex of person on his choice was noted. Brajša-Žganec, Merkaš, and
Šverko (2011) revealed that involvement in leisure activities are main reason of subjective
wellbeing, which also varies across gender and age. Value of leisure was found to be
increasing steadily and value for work centrality declining over the generations (Twenge,
Campbell, Hoffman, & Lance, 2010). There have been literatures citing the effect gender
difference separately for leisure and importance of work (Twenge, 2010; Henderson, &
Gibson, 2013). Difference in importance between work and leisure in life with respect to
gender of people was in addition to these previous literatures.
Appendices - 3

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Side-by-side Box plot for Leisure and Work on sex
Descriptive Statistics

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Dependent ANOVA results
Test for Homogeneity

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Multi Factor ANOVA output
Exercise 4
Hypotheses
Null hypothesis: H0: All three types of reinforcements had equal effect on “children’s
self-esteem”.
Alternate hypothesis: HA: At least one of the three types of reinforcements had
different effect on “children’s self-esteem”.
Results
Variables
DV: “children’s self-esteem” (continuous), IVs: Types of reinforcements (categorical)
Choice of test
One way independent ANOVA (as impact of one categorical variable (Types of
reinforcements) on a continuous variable (“children’s self-esteem”) was checked).
Testing of Assumptions
Each sample observation of child’s self-esteem was independent random samples.
Normality of “children’s self-esteem” for three types of reinforcements (Competence
(W(20) = 0.94, p = 0.20), Agility (W(20) = 0.96, p = 0.64), and Creativity (W(20) = 0.94, p
= 0.21)) was obtained from Shapiro Wilk test for normality.

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Test of homogeneity from Levene’s test revealed that variances for “children’s self-
esteem” for three types of reinforcements were statistically equivalent (F(2,57) = 0.26, p =
0.77).
One outlier observation each for Competence and Agility were noted during checking
of normality with side by side box plots. Assumption was just violated, but the testing
continued with ANOVA.
Findings
Average scores for “children’s self-esteem” for three types of reinforcements, and
their respective standard deviations have been provided in Table 3.
Table 3: Average and Standard Deviation of “children’s self-esteem” for three types of reinforcements
“children’s self-
esteem” N M SD
Competence 20 38.35 3.87
Agility 20 19.8 4.549
Creativity 20 15.1 3.972
Total 60 24.42 10.91
A one-way ANOVA was used for testing whether all three types of reinforcements
had equal effect on “children’s self-esteem”. A significant impact of three types of
reinforcements was observed ( F ( 2 , 57 ) =176 . 24 , p<0 . 01 ) at 1% level of significance.
Pairwise comparison in post-hoc analysis with Tukey’s test indicated that significant
differences existed between Competence and Agility ( p<0 . 01 ) , Competence and Creativity
( p<0 . 01 ) , and Creativity and Agility ( p<0 . 01 ) . Hence, Competence had the best impact on
“children’s self-esteem”, followed by Agility, and then Creativity.
Figure 2: Means Plot for Children Self-Esteem on three types of reinforcements

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Discussion
Self-esteem for each child completing three puzzles and receiving complements was
calculated based on Competence, Agility or Creativity. Self-esteem scores were found to be
unalike for three types of reinforcement. It was also noted that Competence had the best
impact on “children’s self-esteem”, followed by Agility, and then Creativity. Self-esteem in
children is an important aspect of their proper mental and physical growth. It is known as the
feeling of self-appreciation and is essential to adapt to society (Hosogi, Okada, Fujii,
Noguchi, & Watanabe, 2012). Previous literatures on agility and resilience has emphasized
on social relationships and effective schools for children’s strength, self-esteem and healing
from the natural world (Chawla, 2014). A comparative study with all three essential
components of self-esteem for a child was first of its kind. The present research emphasized
from the angle of relative importance of three reinforcements in a child’s growing age. Future
study with large sample size and age, and gender of children would enable to draw a
generalized conclusion.
Appendices – 4
Shapiro-Wilk test for Normality

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Descriptive Statistics of self-esteem on type of reinforcements
One-Way independent ANOVA results

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Side-by-side Box plot for self-esteem on reinforcements

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References
Brajša-Žganec, A., Merkaš, M., & Šverko, I. (2011). Quality of life and leisure activities:
How do leisure activities contribute to subjective well-being?. Social Indicators
Research, 102(1), 81-91.
Chawla, L. (2014). Children’s engagement with the natural world as a ground for healing. In
Greening in the red zone (pp. 111-124). Springer, Dordrecht.
Delhey, J. (2010). From materialist to post-materialist happiness? National affluence and
determinants of “life satisfaction” in cross-national perspective. Social Indicators
Research, 97(1), 65-84.
Elgar, F. J., Davis, C. G., Wohl, M. J., Trites, S. J., Zelenski, J. M., & Martin, M. S. (2011).
Social capital, health and “life satisfaction” in 50 countries. Health & place, 17(5),
1044-1053.
Henderson, K. A., & Gibson, H. J. (2013). An integrative review of women, gender, and
leisure: Increasing complexities. Journal of Leisure Research, 45(2), 115-135.
Hosogi, M., Okada, A., Fujii, C., Noguchi, K., & Watanabe, K. (2012). Importance and
usefulness of evaluating self-esteem in children. BioPsychoSocial medicine, 6(1), 9.
Inglehart, R. (2010). Faith and freedom: Traditional and modern ways to happiness.
International differences in well-being, 351-397.
Okulicz-Kozaryn, A. (2010). Religiosity and “life satisfaction” across nations. Mental
Health, Religion & Culture, 13(2), 155-169.
Pénard, T., Poussing, N., & Suire, R. (2013). Does the Internet make people happier?. The
Journal of Socio-Economics, 46, 105-116.
Senik, C. (2011). Is happiness different from flourishing? Cross-country evidence from the
ESS. Revue d'économie politique, 121(1), 17-34.
Twenge, J. M. (2010). A review of the empirical evidence on generational differences in
work attitudes. Journal of Business and Psychology, 25(2), 201-210.
Twenge, J. M., Campbell, S. M., Hoffman, B. J., & Lance, C. E. (2010). Generational
differences in work values: Leisure and extrinsic values increasing, social and
intrinsic values decreasing. Journal of management, 36(5), 1117-1142.
Welzel, C., & Inglehart, R. (2010). Agency, values, and well-being: A human development
model. Social indicators research, 97(1), 43-63.

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Ye, S., Yu, L., & Li, K. K. (2012). A cross-lagged model of self-esteem and “life
satisfaction”: Gender differences among Chinese university students. Personality and
Individual Differences, 52(4), 546-551.