---
title: "Normal confidence intervals"
author: "Milica Cudina"
date: " "
output: pdf_document
---

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

First, I will set up the **confidence level**.
```{r}
c=0.80
```
The critical value associated with this confidence level is
```{r}
z.star=qnorm((1+c)/2)
z.star
```

Next, let's draw a sample of size $n=100$ from the standard normal distribution.
```{r}
n=100
srs=rnorm(n,0,1)
srs
```
What's the value of the point estimate for the mean in this sample?
```{r}
x.bar=mean(srs)
x.bar
```
Assume that the $\sigma_X$ parameter is known. What's the value of the standard error? 
```{r}
sigma=1
s.e=sigma/sqrt(n)
s.e
```
What's the value of the margin of error?
```{r}
m.e=z.star*s.e
m.e
```
Then the confidence interval will be $\bar x \pm z^*\left(\frac{\sigma}{\sqrt{n}}\right)$. Is the theoretical mean in the confidence interval?
```{r}
abs(x.bar-0)<m.e
```
What if we repeat this procedure a number of times? Say, 1000?
```{r}
n=100
n.sim=1000
in.conf.int=numeric(0)
for (i in 1:n.sim){
  srs=rnorm(n,0,1)
  x.bar=mean(srs)
  in.conf.int=append(in.conf.int,as.numeric(abs(x.bar-0)<m.e))
}
mean(in.conf.int)
```