Part 1

Synopsis The project consists of two parts: 1. Simulation Exercise to explore inference 2. Basic inferential analysis using the ToothGrowth data in the R datasets package

Part 1: Simulation Exercise

The task is to investigate the exponential distribution in R and compare it with the Central Limit Theorem. The exponential distribution will be simulated in R with rexp(n,lambda) where lambda is the rate parameter. The mean of exponential distribution and the standard deviation are both 1/lambda where lambda = 0.2, and distribution of averages of 40 exponentials and will perform 1000 simulations.

library(ggplot2)

## Warning: package 'ggplot2' was built under R version 4.3.1

library(plyr)

## Warning: package 'plyr' was built under R version 4.3.2

library(datasets)
library(tinytex)


set.seed(2000)
lambda <- 0.2
n <- 40
sim_data <- replicate(1000, rexp(n, lambda))
mean_sim_data <- apply(sim_data, 2, mean)

Mean Comparision Sample Mean vs Theoretical Mean of the Distribution

# Sample Mean
sampleMean <- mean(mean_sim_data) # Mean of sample means
print (paste("Sample Mean = ", sampleMean))

## [1] "Sample Mean =  5.0294103260061"

# Theoretical Mean
# the expected mean of the exponential distribution of rate = 1/lambda
theoretical_mean <- (1/lambda)
print (paste("Theoretical Mean = ", theoretical_mean))

## [1] "Theoretical Mean =  5"

# Histogram shows differences
hist(mean_sim_data, col="light blue", xlab = "Mean Average", main="Distribution of Exponential Average")
abline(v = theoretical_mean, col="brown")
abline(v = sampleMean, col="green")

Question 2: Show how variable the sample is (via variance) and compare it to the theoretical variance of the distribution

Calculating the theoretical and sample variance

# sample deviation & variance
sample_dev <- sd(mean_sim_data)
sample_dev

## [1] 0.7853714

sample_variance <- sample_dev^2
sample_variance

## [1] 0.6168082

# theoretical deviation & variance
theoretical_dev <- (1/lambda)/sqrt(n)
theoretical_dev

## [1] 0.7905694

theoretical_variance <- ((1/lambda)*(1/sqrt(n)))^2
theoretical_variance

## [1] 0.625

Question 3: Show that the distribution is approximately normal Histogram with Density and sample means:

d <- data.frame(mean_sim_data)
t <- data.frame(theoretical_mean)
g <- ggplot(d, aes(x = mean_sim_data)) +

 geom_histogram(binwidth = 0.2, color="black", fill="brown" , aes(y=..density..))+
            stat_function(fun=dnorm, args=list(mean=theoretical_mean, sd=sd(mean_sim_data)), 
                          color="green", size =1) +
            stat_density(geom = "line", color = "blue", size =1)  +
            labs(x="Mean", y= "Density", 
                    title="Normal Distribution Comparision")

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

g

## Warning: The dot-dot notation (`..density..`) was deprecated in ggplot2 3.4.0.
## ℹ Please use `after_stat(density)` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

The above plot indicated that density curve is similar to normal distribution curve.

Q-Q Normal Plot also indicates the normal distribution

qqnorm(mean_sim_data)
 qqline(mean_sim_data, col = "magenta")