JHS Statistical Inference Final Project part 1.Rmd

---
title: "JHS statistical inference final project Part 1"
author: Yuncheng YANG
output:
  html_document:
    df_print: paged
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

# Simulation Exercise
This exercise will simulate random exponetial distribution of 1000 trials with sample size of 40 
The mean and variance of sample will be used to explore central limited theorm

### Show the sample mean and compare it to the theoretical mean of the distribution
Accroding to the central limit therom, the mean of sample mean distribution can estimate the population mean
Here, we set population mean should be in 95% condifence level of sample mean distribution
```{r, echo=FALSE} 
lambda<-0.2
#simulating 1000 trials,sample size of 40
n=40
rows=1000
sim<-rexp(n*1000, lambda)
##calculating sample means 
sim1<-matrix(sim,rows)
smean1<-rowMeans(sim1)
hist(smean1)
```
Now use ggplot2 package to explore and visualize the data better and show approximately normal distribution
```{r, echo=FALSE}
library(ggplot2)
smean1<-data.frame(smean1)
g<-ggplot(smean1, aes(x=smean1))+geom_density(fill="lightgreen", color="white")
##showing normal distribution and plotting the vertical line of sample mean
g+stat_function(fun = dnorm,args = list(mean = 1/lambda, sd = 1))+geom_vline(xintercept = mean(smean1$smean1), color="blue", size=2)

```

From the plot, we can see the mean of sample mean distribution is merely population mean(1/0.2=5)
Now we perform t test
```{r, echo=TRUE}
t.test(smean1$smean1,lower.tail=F)
```

95% confidence interval included the population mean, p-value< 2.2e-16, hence it is approximately normal distribution,CLT applied in this case.

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

```{r, echo=TRUE}
#calculate sample variance and substract population variance and plot the distribution
svar<-apply(sim1,1,var)
svar<-data.frame(1:1000, svar)
g1<-ggplot(svar, aes(x=svar))+geom_histogram(fill="lightblue", color="white")
g1 +geom_vline(xintercept = (1/lambda)^2, color="blue", size=2)

```
The distribution still appeared to be a bell shape but more skewed

```{r, echo=TRUE}
##perfoming 2 headed t.test
t.test(svar$svar,lower.tail=F)
```

p value <2.2e-16, for 95% confidence level between 24.82609 and 26.31739, which included population mean