author：huxinyin

Abstract

This report presents a simulation exercise comparing the distribution of sample means and variances to their theoretical counterparts. It also includes a basic inferential data analysis of the ToothGrowth dataset.

Part 1: Simulation Exercise

# Load necessary libraries
library(ggplot2)
## Warning: 程序包'ggplot2'是用R版本4.4.3 来建造的
# Set lambda to 0.2
lambda <- 0.2

# Simulate the distribution of the mean of 40 exponential random variables
set.seed(123)  # Set a random seed for reproducibility
mns <- numeric(1000)
for (i in 1:1000) {
  mns[i] <- mean(rexp(40, lambda))
}

# Plot the distribution of sample means
hist(mns, main="Distribution of Sample Means", xlab="Sample Mean", col="lightblue", border="black")
abline(v=1/lambda, col="red", lwd=2)  # Theoretical mean

# Save the plot
ggsave("sample_means_distribution.png")
## Saving 7 x 5 in image
# Calculate the sample variances
vars <- numeric(1000)
for (i in 1:1000) {
  vars[i] <- var(rexp(40, lambda))
}

# Plot the distribution of sample variances
hist(vars, main="Distribution of Sample Variances", xlab="Sample Variance", col="lightgreen", border="black")
abline(v=(1/lambda)^2, col="red", lwd=2)  # Theoretical variance

# Save the plot
ggsave("sample_variances_distribution.png")
## Saving 7 x 5 in image
# Display the comparison of sample mean and theoretical mean
print(paste("Sample Mean:", mean(mns)))
## [1] "Sample Mean: 5.01191127862178"
print(paste("Theoretical Mean:", 1/lambda))
## [1] "Theoretical Mean: 5"
# Display the comparison of sample variance and theoretical variance
print(paste("Sample Variance:", mean(vars)))
## [1] "Sample Variance: 24.7931154965223"
print(paste("Theoretical Variance:", (1/lambda)^2))
## [1] "Theoretical Variance: 25"

Part 2: Basic Inferential Data Analysis

# Load the ToothGrowth dataset
data("ToothGrowth")

# Provide a basic summary of the data
summary(ToothGrowth)
##       len        supp         dose      
##  Min.   : 4.20   OJ:30   Min.   :0.500  
##  1st Qu.:13.07   VC:30   1st Qu.:0.500  
##  Median :19.25           Median :1.000  
##  Mean   :18.81           Mean   :1.167  
##  3rd Qu.:25.27           3rd Qu.:2.000  
##  Max.   :33.90           Max.   :2.000
# Basic exploratory data analysis
plot(ToothGrowth$len, ToothGrowth$dose, col=ToothGrowth$supp, pch=19, xlab="Dose", ylab="Tooth Length")
legend("topright", legend=levels(ToothGrowth$supp), col=1:2, pch=19)

# Save the plot
ggsave("toothgrowth_scatterplot.png")
## Saving 7 x 5 in image
# Analyze the data using a linear model
model <- lm(len ~ dose + supp, data=ToothGrowth)
summary(model)
## 
## Call:
## lm(formula = len ~ dose + supp, data = ToothGrowth)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -6.600 -3.700  0.373  2.116  8.800 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   9.2725     1.2824   7.231 1.31e-09 ***
## dose          9.7636     0.8768  11.135 6.31e-16 ***
## suppVC       -3.7000     1.0936  -3.383   0.0013 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.236 on 57 degrees of freedom
## Multiple R-squared:  0.7038, Adjusted R-squared:  0.6934 
## F-statistic: 67.72 on 2 and 57 DF,  p-value: 8.716e-16
# Confidence intervals
confint(model)
##                 2.5 %    97.5 %
## (Intercept)  6.704608 11.840392
## dose         8.007741 11.519402
## suppVC      -5.889905 -1.510095
# Hypothesis testing
t.test(len ~ supp, data=ToothGrowth)
## 
##  Welch Two Sample t-test
## 
## data:  len by supp
## t = 1.9153, df = 55.309, p-value = 0.06063
## alternative hypothesis: true difference in means between group OJ and group VC is not equal to 0
## 95 percent confidence interval:
##  -0.1710156  7.5710156
## sample estimates:
## mean in group OJ mean in group VC 
##         20.66333         16.96333