---
title: "Homework assignment #10"
author: "Milica Cudina"
date: "`r Sys.Date()`"
output: pdf_document
---
<!-- The author of this template is Dr. Gordan Zitkovic.-->
<!-- The code chunk below contains some settings that will  -->
<!-- make your R code look better in the output pdf file.  -->
<!-- If you are curious about what each option below does, -->
<!-- go to https://yihui.org/knitr/options/ -->
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```{r echo=FALSE, warning=FALSE, include=FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  fig.align="center",
  fig.pos="t",
  strip.white = TRUE,
  tidy = TRUE
)
```
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## Problem #1 (10 points)
You are rolling a fair dodecahedral ($20-$sided) die with numbers $1-20$ noted its sides. You win if the number rolled is prime and you lose if it is not. What is the probability that you win at most $200$ times in $500$ rolls? 
Calculate your answer \underline{\bf both}:
\begin{itemize}
  \item[(i)] ($4$ points) using {\bf R} without an approximation, and
  \item[(ii)] ($6$ points) using the normal approximation. 
\end{itemize}


## Problem #2 (10 points)
A biased coin (probability of {\em heads} is 0.7) 
is tossed 1000 times. Write down the exact expression for the
probability that more than 750 {\em heads} have been observed. 
\begin{itemize}
  \item[(i)] ($4$ points) Use {\bf R} to calculate this probability.
  \item[(ii)] ($6$ points) Use the normal approximation to approximate this probability.
\end{itemize}


## Problem #3 (2+2+4+1+1=10 points)
Solve **Exercise 6.8** from the textbook. 


## Problem #4 ($4+4=8$ points)
Solve **Exercise 6.14** from the textbook. 


## Problem #5 ($4$ points)
Solve **Exercise 6.16** from the textbook. 


## Problem #6 ($4$ points)
In a random sample of $1000$ small children, it was found that $880$ of them observe Halloween. Find the 80\%-confidence interval for the population proportion of children who observe Halloween.   


## Problem #7 ($4$ points)
Let $p$ denote the population proportion. How large should the sample size be so that one is at least 95\% confident that the true parameter $p$ is within a 0.02 margin of error from the point estimate?