---
title: "SML201 Precept 8, Spring 2020"
output: html_document
---

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

### Problem 1

Suppose that $X\sim\mathcal{N}(2, 10^2)$. We sample the variable $X$ once (i.e., we obtain a sample from the distribution $\mathcal{N}(2, 10^2)$). In this problem, you will be computing the same quantity in four different ways. You should expect to get roughly the same answer every time.

### Problem 1(a)

Write R code to obtain $P(2.1 < X < 3.1)$. Use `pnorm`.

### Problem 1(b)

Write R code to obtain $P(2.1 < X < 3.1)$. Use `pnorm(..., ,mean = 0, sd = 1)`.

### Problem 1(c)

Write R code to obtain $P(2.1 < X < 3.1)$. Use `rnorm`. (And not `pnorm`.)

### Problem 1(d)

Write R code to obtain $P(2.1 < X < 3.1)$. Use `rnorm(..., mean = 0, sd = 1)`


### Problem 2
Suppose 65% of Princeton students like Wawa better than World Coffee. We selected a random sample of 100 students, and asked them which they prefer. What is the probability that more than 78 students said "Wawa"? 

#### Problem 2(a)

Answer the question using `pbinom`.


#### Problem 2(b)

Answer the question using `pnorm`. Use the normal approximation to the Binomial distribution (recall: the mean is $n\times prob$ and the variance is $n\times prob\times (1-prob)$). 

(Note: you shouldn't expect an exact match between 2(a) and 2(b) because of the lack of continnuity correction. You can try obtaining an exact match by varying the value of the `q`).

### Problem 3

Suppose 100 Princeton students we asked whether Harvard or Stanford is the worse online institution of higher learning. 60 students said that Stanford is worse. Compute the p-value for the null hypothesis that Princeton students think that Harvard and Stanford are equally bad, on average. What can you conclude?

### Problem 4

Answer Problem 2 using only `rnorm(..., mean = 0, sd = 1)`


