Comprehensive Statistics Report: Data Analysis and Covariance

This statistics report provides a detailed analysis of a dataset using various statistical techniques. It begins with exploratory data analysis, examining the distribution of variables such as gender, classroom size, and math scores, including profile plots and descriptive statistics. The report then delves into factorial ANOVA to assess the significance of gender and classroom size on math scores, including post-hoc tests and multiple comparisons. Finally, it covers analysis of covariance (ANCOVA) to understand the impact of gender and training class on performance scores, including tests for homogeneity and pairwise comparisons. The analysis includes interpretations of SPSS outputs and hypothesis testing, offering insights into the relationships between variables and their statistical significance. Desklib offers more solved assignments and past papers for students.

Running head: STATISTICS 2
Statistics 2
Name of the student
Name of the university
Author’s note

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Table of Contents
1. Exploratory Data analysis:.......................................................................................................2
Part a)...........................................................................................................................................2
Part b)...........................................................................................................................................4
Part c)...........................................................................................................................................5
2. Factorial ANOVA (analysis of variance):................................................................................5
Part a)...........................................................................................................................................6
Part b)...........................................................................................................................................6
Part c)...........................................................................................................................................8
Part d)...........................................................................................................................................9
Part e).........................................................................................................................................11
3. Analysis of covariance (ANCOVA):.....................................................................................11
Interpretation of Mock ANCOVA output table:........................................................................16
References:....................................................................................................................................18

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1. Exploratory Data analysis:
In the given data set, mainly three variables are undertaken that are “Gender”, “Classroom”
and “Math_Score”. Among 60 participants, 30 are males and rest of 30 are females. Total 60
students attend their classrooms in three classes with 20 students in each class. There are total six
groups present in the data set according to the paired values of “Gender” and “Classroom”.
The two types of “Gender” are “Male” (M) and “Female” (F). The ordinal variable
“Classroom” is classified in three kinds. “1” refers Small classroom where not more than 10
children could be accommodated. “2” refers Medium classroom where the accommodation is for
11 to 19 students and “3” refers Large classroom where the accommodation is for more than 20
students.
Part a)
Table 1: Table of Between-Subjects Factors
Table 2: Exploratory data summary of Mathematics Score

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Figure 1: Profile plot of Estimated Marginal Means of Scores of Mathematics with respect
to Gender

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Figure 2: Profile plot of Estimated Marginal Means of Scores of Mathematics with respect
to Classroom
Part b)
The SPSS generated profile plot shows that both males and females average score is
highest in mathematics in case of small classroom. For the females, the average score in
mathematics decreases rapidly as the size of classroom increases. However, for the males, the
average score of mathematics first decreases as classroom size turns out to be small to medium
and then increases as the size of classroom becomes Large. The difference of average scores of
mathematics for male and female in small and medium classroom are not high. However, the

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difference of average scores of mathematics for male and female in large classroom is
significantly high.
From the other perspective, the average score of mathematics in small classroom is higher for
females than males. However, Males score in mathematics more than Females score in medium
and large size classroom.
Part c)
Table 3: Descriptive Statistics of Mathematics Score
The average Scores in mathematics for the accounted six groups are –
a) Mean scores of “Females” whose classroom was “small” = 93.8
b) Mean scores of “Females” whose classroom was “medium” = 88.5
c) Mean scores of “Females” whose classroom was ‘large” = 79.2

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d) Mean scores of “Males” whose classroom was “small” = 92.7
e) Mean scores of “Males” whose classroom was “medium” = 89.7
f) Mean scores of “Males” whose classroom was “large” = 91.2
2. Factorial ANOVA (analysis of variance):
Table 4: Tests of Between-Subjects Effects in Factorial ANOVA method
Part a)
According to the Factorial ANOVA table, the main effect of the factor “Gender” has p-
value 0.0 (<0.05). Hence, the effect of “Gender” is insignificant. It could be inferred that the
average scores in mathematics in both kinds of “Gender” are unequal. The main effect of gender
is found statistically significant in this analysis.
Part b)
The p-value of main effect of the factor “Classroom” has p-value 0.0 (<0.05). Hence, the
effect of “Classroom” is also insignificant. We can conclude that the average scores in

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mathematics for all the three types of classroom sizes are not equal (Source:
Academic.udayton.edu., 2018). The main effect of classroom size is also found statistically
significant in this analysis.
Table 5: Table of Estimated Marginal means of Gender
Overall, the average score of mathematics in case of males is greater than females (91.2>87.167).
Table 6: Table of Estimated Marginal means of Classroom
The students of small classroom scores is greater than the scores of students of medium
classroom (93.25>89.1). The scores in mathematics are least for the students of large classroom
(85.2).
Table 7: Table of Estimated Marginal means of interaction effect of Gender and Classroom

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Female students who get learning in small classroom generally get maximum score in
mathematics. On the other hand, female students who get learning in large classroom, score least
marks in the mathematics exam.
Table 8: Table of Post Hoc test of Multiple Comparisons of Classrooms

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Part c)
The previous two-way ANOVA table indicates p-value of interaction effect of “Gender”
and “Classroom” as 0.0 (<0.05). Therefore, the interaction of these two factors has statistical
significance. It could be concluded that the effect of types of gender and classrooms are not
same. Here, the interaction effect of these two factors is statistically significant too.
The difference average scores in mathematics of the small and large classroom students
are significantly high (±8.05) followed by small and medium classroom students (±4.15). All the
p-values (0.0, 0.002 and 0.003) of differences of mean score are less than 0.05
(Statistics.laerd.com., 2018). Therefore, we can conclude that the differences of average scores in
mathematics are significant to each other.
Table 9: Table of Homogeneous Subsets of the Mathematics Scores
The average scores of mathematics for three sizes of classrooms are undertaken here. The
table of homogeneity provides the significant p-value = 1 (>0.05). Therefore, the assertion of
absence of homogeneity in all the three types of classrooms could be rejected (Lomax and
Surman, 2007).

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Part d)
The hypotheses that are to be tested here are:
Null hypothesis (H0): The girls score higher than boys in the maths exam when small classroom
is accommodated.
Alternative hypothesis (HA): The girls score same as boys in the maths exam when small
classroom is accommodated.
Table 10: Tables for testing the equality of averages of scores of maths for the small
classroom among 10 boys and 10 girls

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The average of score of maths in small classroom is 93.8 for females and 92.7 for males. The
ANOVA table which verifies the equality of averages provides the F-statistic 0.443 with
calculated p-value = 0.514. The p-value is greater than 0.05. Therefore, we accept the null
hypothesis that girls perform better than boys do when capacity of classroom is fewer (small
classroom).
Part e)
According to the two-way factorial ANOVA table as well as the Post-hoc test and multiple
comparison test, it is evident that the both the main effects “Gender” and “Classroom” have
significant associations with score of mathematics. Gender and Classroom in collaborate have
their significant interaction effect that also takes part in the deviation of maths score (Field,
2013).
Simple main effects analysis indicated that-