Homework 2
2025-03-25
Stella Maria Lourdes Kusumawardhani
2) Import data
mydata <- read.table("./student-mat.csv", header = TRUE, sep = ";", dec = ",")library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionmydata <- mydata %>%
select(sex, age, studytime, absences, G1)- due to the excessive amount of variables, 5 variables of interest are chosen from the original dataset
3) Data display
head(mydata)## sex age studytime absences G1
## 1 F 18 2 6 5
## 2 F 17 2 4 5
## 3 F 15 2 10 7
## 4 F 15 3 2 15
## 5 F 16 2 4 6
## 6 M 16 2 10 154) Data explanation
- The processed dataset contains 395 observations and 5 variables
- The unit of observation is an individual student
Variables:
- sex
- sex of student
- type: categorical, nominal
- F : Female
- M : Male
- age
- age of student
- type: numeric, ratio
- unit: years
- studytime
- weekly study time of student
- type: categorical, ordinal
- 1 : less than 2 hours
- 2 : 2 to 5 hours
- 3 : 5 to 10 hours
- 4 : more than 10 hours
- absences
- number of student’s school absences
- type: numeric, ratio
- unit: days
- grade
- student’s mathamatics grade during first evaluation
- type: numeric, ratio
- unit: points
5) Data source
Cortez, P. (2008). Student Performance [Dataset]. UCI Machine Learning Repository. https://doi.org/10.24432/C5TG7T.
6) Data manipulation
mydata <- mydata %>%
rename(gender = sex, grade = G1)mydata$gender <- factor(mydata$gender,
levels = c("F", "M"),
labels = c("Female", "Male"))mydata$studytime <- factor(mydata$studytime,
levels = c(1, 2, 3, 4),
labels = c("< 2 hours", "2 to 5 hours", "5 to 10 hours", "> 10 hours"),
ordered = TRUE)Descriptive Statistics
summary(mydata)## gender age studytime absences
## Female:208 Min. :15.0 < 2 hours :105 Min. : 0.000
## Male :187 1st Qu.:16.0 2 to 5 hours :198 1st Qu.: 0.000
## Median :17.0 5 to 10 hours: 65 Median : 4.000
## Mean :16.7 > 10 hours : 27 Mean : 5.709
## 3rd Qu.:18.0 3rd Qu.: 8.000
## Max. :22.0 Max. :75.000
## grade
## Min. : 3.00
## 1st Qu.: 8.00
## Median :11.00
## Mean :10.91
## 3rd Qu.:13.00
## Max. :19.00Interpretation
- age
- mean: the average age of students is 16.7 years old
- median: half of the students are aged less then or equal to 17 years old, while the other half are aged over 17 years old
- min, max: the youngest student in the sample is aged 15 years old, the oldest student is aged 22 years old
- absences
- mean: the average days of absences of the students is 5.7 days
- median: half of the students were absent for less than or equal to 4 days, while the other half were absent for over 4 days
- min, max: the fewest days for student’s absence is 0 days, the most recorded days for student’s absence is 75 days
- grade
- mean: the average grade of students is 10.91 points
- median: half of the students received a grade of less than or equal to 11 points, while the other half were absent for over 11 points
- min, max: student who received the lowest grade scored 3 points, while student who received the highest grade scored 19 points
library(psych)
describeBy(mydata$grade, group = mydata$gender)##
## Descriptive statistics by group
## group: Female
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 208 10.62 3.23 10 10.46 2.97 4 19 15 0.36 -0.68 0.22
## ------------------------------------------------------------
## group: Male
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 187 11.23 3.39 11 11.19 4.45 3 19 16 0.1 -0.72 0.25**Interpretation:*
- Female
- mean: the average grade for female students is 10.62 points
- median: half of the female students received a grade of less than or equal to 10 points, while the other half received a grade of over 10 points
- min, max: the female student who received the lowest grade scored 4 points, while the female student who received the highest grade scored 19 points
- Male
- mean: the average grade for male students is 11.23 points
- median: half of the male students received a grade of less than or equal to 11 points, while the other half received a grade of over 11 points
- min, max: the male student who received the lowest grade scored 3 points, while the male student who received the highest grade scored 19 points
7) Research Question and Hypothesis Testing
Paremetric Test
Research Question Is there a difference in average mathematics grade between male and female students?
First assumption Grade is a numeric variable
- Assumption fulfilled
Second assumption Grade is normally distributed within the population of male and female students
library(ggplot2)##
## Attaching package: 'ggplot2'## The following objects are masked from 'package:psych':
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## %+%, alphaGrade_Male <- ggplot(mydata[mydata$gender == "Male", ], aes(x = grade)) +
geom_histogram(binwidth = 1, col = "black", fill = "purple") +
theme_linedraw() +
ylab("Points") +
ggtitle("Male Students' Grades")
Grade_Female <- ggplot(mydata[mydata$gender == "Female", ], aes(x = grade)) +
geom_histogram(binwidth = 1, col = "black", fill = "purple") +
theme_linedraw() +
ylab("Points") +
ggtitle("Female Students' Grades")library(ggpubr)
ggarrange(Grade_Male, Grade_Female,
ncol = 2, nrow = 1)
library(ggpubr)
ggqqplot(mydata,
"grade",
facet.by = "gender")
- Normality assumption is **not fulfilled*, grade is not normally distributed within the Female students population
Third Assumption Grade has the same variance in both male and female students population
library(car)## Loading required package: carData##
## Attaching package: 'car'## The following object is masked from 'package:psych':
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## logit## The following object is masked from 'package:dplyr':
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## recodeleveneTest(mydata$grade, group = mydata$gender)## Levene's Test for Homogeneity of Variance (center = median)
## Df F value Pr(>F)
## group 1 0.4903 0.4842
## 393- the variance for grade is the same in both population
- third assumption is fulfilled
t.test(mydata$grade ~ mydata$gender,
var.equal = TRUE,
alternative = "two.sided")##
## Two Sample t-test
##
## data: mydata$grade by mydata$gender
## t = -1.8284, df = 393, p-value = 0.06825
## alternative hypothesis: true difference in means between group Female and group Male is not equal to 0
## 95 percent confidence interval:
## -1.26541466 0.04590623
## sample estimates:
## mean in group Female mean in group Male
## 10.62019 11.22995library(effectsize)##
## Attaching package: 'effectsize'## The following object is masked from 'package:psych':
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## phieffectsize::cohens_d(mydata$grade ~ mydata$gender,
pooled_sd = FALSE)## Cohen's d | 95% CI
## -------------------------
## -0.18 | [-0.38, 0.01]
##
## - Estimated using un-pooled SD.interpret_cohens_d(0.18, rules = "sawilowsky2009")## [1] "very small"
## (Rules: sawilowsky2009)Results - Based on the sample data, we fail to reject the null hypothesis that the average mathematics grade of male and female students does not differ (p = 0.07).The effect size is very small (d = 0.18)
Non-Parametric Test
Research Question Is there a difference in mathematics grade between male and female students?
wilcox.test(mydata$grade ~ mydata$gender,
correct = FALSE,
exact = FALSE,
alternative = "two.sided")##
## Wilcoxon rank sum test
##
## data: mydata$grade by mydata$gender
## W = 17322, p-value = 0.05951
## alternative hypothesis: true location shift is not equal to 0library(effectsize)
effectsize(wilcox.test(mydata$grade ~ mydata$gender,
correct = FALSE,
exact = FALSE,
alternative = "two.sided"))## r (rank biserial) | 95% CI
## ---------------------------------
## -0.11 | [-0.22, 0.00]interpret_rank_biserial(0.11)## [1] "small"
## (Rules: funder2019)Results - Based on the sample data, we find that the mathematics grade for male and female students do not differ (p = 0.06). The effect size is small (d = 0.11)
Conclusion
- Non-parametric test is more suitable in this particular study, as not all assumptions are met, specifically the normality assumption