---
title: "Quiz #7"
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  -->
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```{r echo=FALSE, warning=FALSE, include=FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  fig.align="center",
  fig.pos="t",
  fig.width=12,
  strip.white = TRUE
)
```
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## Problem 1. **(1 point)** 
The mean and median of any normal distribution are equal. *True or false? Why?*


## Problem 2. **(3 points)** 
Let the population distribution be normal with mean $\mu$ and standard deviation $\sigma$. Let $\bar X$ denote the sample mean of a sample of size $n$from this population. Then, we know the following about the distribution of $\bar X$:

**a.** $\bar X \sim Normal(mean=\mu, variance=\sigma^2)$

**b.** $\bar X \sim Normal(mean=\mu, variance=\tfrac{\sigma^2}{n})$

**c.** $\bar X \sim Normal(mean=\mu, variance=\tfrac{\sigma^2}{\sqrt{n}})$

**d.** $\bar X \sim Normal(mean=\frac{\mu}{n}, variance=\tfrac{\sigma^2}{n})$

**e.** None of the above are correct. 


## Problem 3. **(1 point)**
Suppose a poll suggested the US President’s approval rating is 45\%. We would consider 45\% to be \dots

**a.** the population mean.

**b.** the point estimate.

**c.** the statistic.


## Problem #4 ($5$ points) 
Let $Z \sim N(0,1).$ Given that $Z > 0$, find the probability that $Z < 2.$


## Problem #5 ($5$ points) 
Let the monthly profit of a local cupcakery be normally distributed with mean 
\$20,000 and standard deviation of \$4,000. What is the probability that the combined profit in the months of October and November exceeds \$36,000 (assuming that profits over different months are independent)?