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Case Study 4: Statistical Analysis

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Introduction

In this case study, we will apply the statistical analysis techniques learned in Module 4 to a real-world dataset. The goal is to perform a comprehensive statistical analysis, including descriptive statistics, hypothesis testing, correlation, regression, and ANOVA. This will help solidify your understanding of these concepts and demonstrate their practical applications.

Dataset Description

For this case study, we will use a dataset containing information about students' performance in exams. The dataset includes the following variables:

  • student_id: Unique identifier for each student
  • gender: Gender of the student (Male/Female)
  • math_score: Score in the math exam
  • reading_score: Score in the reading exam
  • writing_score: Score in the writing exam
  • study_hours: Number of hours spent studying per week
  • parental_education: Highest level of education attained by the student's parents

Step-by-Step Analysis

  1. Descriptive Statistics

First, we will calculate the basic descriptive statistics for the dataset to understand the distribution of the variables.

# Load necessary libraries
library(dplyr)

# Load the dataset
data <- read.csv("student_performance.csv")

# Calculate descriptive statistics
summary(data)

Explanation:

  • summary(data) provides a quick overview of the dataset, including the minimum, maximum, mean, and quartiles for each numeric variable.

  1. Hypothesis Testing

Next, we will perform a hypothesis test to determine if there is a significant difference in math scores between male and female students.

# Perform t-test
t_test_result <- t.test(math_score ~ gender, data = data)

# Print the result
print(t_test_result)

Explanation:

  • The t.test() function is used to perform an independent t-test to compare the means of math scores between male and female students.

  1. Correlation Analysis

We will calculate the correlation matrix to examine the relationships between the numeric variables in the dataset.

# Calculate correlation matrix
cor_matrix <- cor(data %>% select(math_score, reading_score, writing_score, study_hours))

# Print the correlation matrix
print(cor_matrix)

Explanation:

  • The cor() function calculates the correlation matrix for the selected numeric variables, showing the strength and direction of the relationships between them.

  1. Regression Analysis

We will perform a linear regression analysis to predict math scores based on study hours and parental education level.

# Convert parental education to a factor
data$parental_education <- as.factor(data$parental_education)

# Fit the linear regression model
reg_model <- lm(math_score ~ study_hours + parental_education, data = data)

# Print the summary of the model
summary(reg_model)

Explanation:

  • The lm() function fits a linear regression model to predict math_score using study_hours and parental_education as predictors.
  • The summary() function provides detailed information about the model, including coefficients, R-squared value, and p-values.

  1. ANOVA

Finally, we will perform an ANOVA to determine if there are significant differences in math scores based on parental education levels.

# Perform ANOVA
anova_result <- aov(math_score ~ parental_education, data = data)

# Print the summary of the ANOVA
summary(anova_result)

Explanation:

  • The aov() function performs an analysis of variance (ANOVA) to test if there are significant differences in math_score across different levels of parental_education.
  • The summary() function provides the ANOVA table with F-values and p-values.

Practical Exercises

Exercise 1: Descriptive Statistics

Calculate the mean, median, and standard deviation for the reading_score variable.

Solution:

mean_reading <- mean(data$reading_score)
median_reading <- median(data$reading_score)
sd_reading <- sd(data$reading_score)

mean_reading
median_reading
sd_reading

Exercise 2: Hypothesis Testing

Perform a hypothesis test to determine if there is a significant difference in writing scores between male and female students.

Solution:

t_test_writing <- t.test(writing_score ~ gender, data = data)
print(t_test_writing)

Exercise 3: Correlation Analysis

Calculate the correlation between math_score and study_hours.

Solution:

cor_math_study <- cor(data$math_score, data$study_hours)
cor_math_study

Exercise 4: Regression Analysis

Fit a linear regression model to predict writing_score based on reading_score and study_hours.

Solution:

reg_model_writing <- lm(writing_score ~ reading_score + study_hours, data = data)
summary(reg_model_writing)

Exercise 5: ANOVA

Perform an ANOVA to determine if there are significant differences in reading_score based on parental_education levels.

Solution:

anova_reading <- aov(reading_score ~ parental_education, data = data)
summary(anova_reading)

Conclusion

In this case study, we applied various statistical analysis techniques to a dataset of student performance. We calculated descriptive statistics, performed hypothesis testing, examined correlations, conducted regression analysis, and performed ANOVA. These exercises reinforced the concepts learned in Module 4 and demonstrated their practical applications. By completing this case study, you should now have a solid understanding of how to perform statistical analysis in R.

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