#### 1. You roll five six-sided dice. Write a script in R to calculate the probability of getting between 15 and 20 (inclusive) as the total amount of your roll (ie, the sum when you add up what is showing on all five dice). Exact solutions are preferable but approximate solutions are ok as long as they are precise.

```{r, eval = FALSE}
p <- 8
results <- numeric(10^p)
count <- 0
for (i in 1:10^p) {
result <- sum(sample(6, 5, replace=TRUE))
if (result>=15 & result<=20) {
count <- count + 1
}
}

count/10^p

```
It is about $0.5571231$.

#### 2. Create a simulated dataset of 100 observations, where x is a random normal variable with mean 0 and standard deviation 1, and $y = 0.1 + 2 x + \epsilon$, where epsilon is also a random normal error with mean 0 and sd 1. (One reminder: remember that in creating simulated data with, say, 100 observations, you need to use `rnorm(100)` for epsilon, not `rnorm(1)`, to ensure that each observation gets a different error.)

```{r}
x <- rnorm(100)
epsilon <- rnorm(100)
y <- 0.1 + 2 * x + epsilon
```
##### a. Perform a t test for whether the mean of Y equals the mean of X using $R$.

```{r}
t.test(y, x)

```
##### b. Now perform this test by hand using just the first 5 observations. Please write out all your steps in latex.

- find the means for y and x, and denote them as $\bar{y}$ and $\bar{x}$ respectively:
```{r}
y.bar <- mean(y[1:5])
x.bar <- mean(x[1:5])
```
- Calculate the group standard error using
$$ s.e._{group} = \sqrt{\frac{s_{1}^2}{n_{1}}+\frac{s_{2}^2}{n_{2}}}$$
where $s_{i}$ i=1,2 is a standard deviation for $y$ and $x$ and $n_{i}$ i=1, 2 is the sample size of $y$ and $x$, 5 in this case.
```{r}
sd.group <- sqrt(sd(y)/5 + sd(x)/5)
```

Use R (RStudio Notebook) to analyze and interpret the following dat...