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Hypothesis Testing using Bootstrap Methods, Resampling Techniques, and Confidence Intervals

Background
Conduct hypothesis testing using bootstrap methods, implement resampling techniques, and compute confidence intervals. The assignment will incorporate a project developed in R, a report presenting the results. It will also incorporate a research review on the current state of Bootstrapping techniques utilization in Data Science.

Instructions
Using this dataset contains physicochemical properties and quality ratings of red and white variants of the Portuguese "Vinho Verde" wine. Features include fixed acidity, volatile acidity, citric acid, residual sugar, chlorides, free sulfur dioxide, total sulfur dioxide, density, pH, sulphates, alcohol content, and a final quality rating from 0 (very bad) to 10 (very excellent).

Source: The UCI Machine Learning Repository - Wine Quality Dataset (https://archive.ics.uci.edu/ml/datasets/wine+quality)

Setup and Data Preparation
Install and load necessary R packages: tidyverse for data manipulation and visualization, boot for bootstrap analysis.
Download the Wine Quality Dataset.
Read the data into R using read.csv() and perform initial data exploration with functions like summary() and head().
Exploratory Data Analysis (EDA)
Visualize the distribution of wine quality ratings for both red and white wine samples.
Explore relationships between physicochemical properties and wine quality using scatter plots and correlation analysis.
Formulate a Hypothesis
Example hypothesis: "The average alcohol content of high-quality wine (rating >= 7) is significantly higher than that of lower-quality wine (rating < 7)."
Bootstrap Resampling for Hypothesis Testing
Implement bootstrap resampling to estimate the difference in mean alcohol content between high-quality and low-quality wines.
Draw many resamples with replacement from the observed dataset, compute the mean alcohol content for high-quality and low-quality wines in each resample, and calculate the difference.
Compute Confidence Intervals
Use the bootstrap samples to compute a 95% confidence interval for the mean difference in alcohol content.
Interpret the confidence interval in the context of the hypothesis.
Perform Hypothesis Testing
Determine whether the observed difference in means is statistically significant based on the bootstrap confidence interval.
Discuss the p-value interpretation and whether the null hypothesis can be rejected.

Sample Answer

R Script for Bootstrap Hypothesis Testing (Wine Quality Data)

This script assumes the combined red and white wine quality datasets have been merged and saved into a single CSV file named wine_quality.csv with a column named quality.

1. ⚙️ Setup and Data Preparation

# 1. Install and Load Packages (Run install.packages() if not installed)
# install.packages(c("tidyverse", "boot"))
library(tidyverse)
library(boot)

# 2. Download and Read Data
# NOTE: Replace "path/to/your/wine_quality.csv" with the actual path
# Assuming a merged dataset is used for simplicity, or load red and white separately.
wine_data <- read_csv("wine_quality.csv") # Use read_csv for tidyverse consistency

# 3. Data Preparation for Hypothesis
wine_data <- wine_data %>%
  # Create a binary quality variable: 1 for High Quality, 0 for Low Quality
  mutate(HighQuality = ifelse(quality >= 7, 1, 0))

# 4. Initial Data Exploration
head(wine_data)
summary(wine_data)
# Check the sample sizes for the two groups
wine_data %>% count(HighQuality)

2. 📊 Exploratory Data Analysis (EDA)

Bootstrap Resampling for Hypothesis Testing

Hypothesis: H0:μHigh−μLow=0 (No difference in mean alcohol content).Ha:μHigh−μLow>0 (High-quality wine has significantly higher mean alcohol content).

The bootstrap statistic we are interested in is the difference in the mean alcohol content between the high-quality group and the low-quality group.

# 1. Define the custom function to compute the statistic of interest
# The 'data' argument is the full resampled dataset from the boot function
# The 'indices' argument specifies which rows were sampled
mean_diff_alcohol <- function(data, indices) {
  # Select the resampled data
  d <- data[indices, ]
  
  # Calculate mean alcohol for each quality group in the resample
  mean_high <- d %>% filter(HighQuality == 1) %>% pull(alcohol) %>% mean()
  mean_low  <- d %>% filter(HighQuality == 0) %>% pull(alcohol) %>% mean()
  
  # Return the difference (High - Low)
  return(mean_high - mean_low)
}

# 2. Perform the Bootstrap Resampling
# R = 1000 is common, R = 5000 is often preferred for more stable confidence intervals
set.seed(42) # For reproducibility
boot_results <- boot(
  data = wine_data,
  statistic = mean_diff_alcohol,
  R = 5000 # Number of bootstrap replicates