STATS 250 Lab 06

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# STATS 250 Lab 06
## Simulation
### Nick Seewald<br />nseewald@umich.edu
### Week of 10/05/2020

---

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# Reminders 💡

Your tasks for the week running Friday 10/2 - Friday 10/9:

| Task | Due Date | Submission |
|:-----|:---------|:-----------|
| M-Write 1 Final Revision | Wednesday 10/7| Canvas |
| Homework 5 | Friday 10/9 8AM ET | course.work |
| Lab 5 | Friday 10/9 8AM ET | Canvas |

Stop by office hours! You can attend anyone's -- not just mine!

M-Write office hours schedule on Canvas (see MWrite Info on home page)

---
# Homework 4 Comments

- Statistics is not a branch of math. It is a *mathematical science*.
  - In statistics, it's important that we tie our conclusions back to data.
--

- **context context context context context**
--

- *ALWAYS* put your answer back into the context of the problem. 
  - What does `\(R^2\)` mean in *this* situation?
  - Why is regression useful to address *this* question?
  
<div style="text-align:center;"><img src="https://media.tenor.com/images/1ab88a521fc0da50d609e14eda65c2aa/tenor.gif" width=500 ></img></div>
---
# Learning Objectives

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### Statistical Learning Objectives
1. Explore sample-to-sample variation
1. Investigate probability using long-run proportions
]
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### R Learning Objectives
1. Learn about reproducible randomness by "setting seeds"
1. Functions within functions: `table(sample())`
1. Line plots in R
]

---
# Weekly Advice

- Randomness is **random**: your mileage may vary when you run code inside chunks.
- Check your HTML file before submitting it! You'll notice formatting issues you can easily fix (often by adding blank lines to your Rmd file).

--
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Please try to follow along with this video. **It will help.**
]
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<br>
<div style="text-align:center;"><img src="https://media1.tenor.com/images/b79c5979e1617eb62ac810eb6d5f6cda/tenor.gif?itemid=16115882" height=300></img></div>
]

---

# Vectors (again)

A *character* vector is a vector where the elements are "strings" of text.

```r
x <- c("hi", "this is", "a character vector.", "Are you impressed?")
x
```

```
[1] "hi"                  "this is"             "a character vector."
[4] "Are you impressed?" 
```

Again, note the use of the `c()` function.

---
# `rep()`

What's easier to code?

```r
pets <- c("cat", "cat", "cat", "cat")
pets
```

```
[1] "cat" "cat" "cat" "cat"
```
--

```r
cats <- rep("cat", 4)
cats
```

```
[1] "cat" "cat" "cat" "cat"
```

`rep(`what you want to repeat`, `number of times to repeat it`)`

---
# `rep()`

What's easier to code?

```r
pets <- c("cat", "cat", "cat", "cat", "dog", "dog", "dog", "dog", "dog")
pets
```

```
[1] "cat" "cat" "cat" "cat" "dog" "dog" "dog" "dog" "dog"
```

```r
pets2 <- c(rep("cat", 4), rep("dog", 5))
pets2
```

```
[1] "cat" "cat" "cat" "cat" "dog" "dog" "dog" "dog" "dog"
```

---
# Functions in Functions
Arguments to functions can be functions! This is called **nesting**.

.pull-left[

```r
table(
  c(
    rep("heads", 5000),
    rep("tails", 5000)
  )
)
```

```

heads tails 
 5000  5000 
```
]
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.center[
![](https://media.tenor.com/images/ae1204bbe4d4cd3e3929990c8c4ed103/tenor.gif)
]
]

**WATCH OUT FOR PARENTHESES**

---
# Remember `sample()`?
We used `sample()` to simulate rolling a die using the vector `1:6`.

We can also give `sample()` a character vector to sample from!

```r
coin <- c('heads', 'tails')
sample(coin, size = 30, replace = TRUE)
```

```
 [1] "heads" "heads" "heads" "heads" "tails" "heads" "tails" "tails" "tails"
[10] "heads" "heads" "tails" "tails" "tails" "tails" "heads" "heads" "heads"
[19] "heads" "heads" "tails" "tails" "heads" "heads" "heads" "tails" "tails"
[28] "tails" "tails" "tails"
```

---
# The `prob` argument to `sample()`

We can simulate a *biased* coin using the `prob` argument.

- `prob` takes a vector of "probability weights", one per element of the vector to sample from
- `prob` applies the weights *in order*

```r
coin <- c('heads', 'tails')
sample(coin, size = 30, replace = TRUE, prob = c(0.3, 0.7))
```

```
 [1] "heads" "tails" "tails" "tails" "heads" "tails" "tails" "tails" "tails"
[10] "tails" "heads" "tails" "tails" "tails" "heads" "heads" "tails" "tails"
[19] "tails" "tails" "tails" "tails" "heads" "heads" "tails" "tails" "tails"
[28] "tails" "tails" "tails"
```

---
# Pseudo-random numbers

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- Humans are very bad at generating random numbers.
- Computers only **seem** better.
- Computers produce *pseudo-random* numbers: if you know the "seed", you know the entire sequence of "random" numbers.
]
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![](random-numbers.png)
]

---
# `set.seed()`

- We can tell R to use a particular "seed" with `set.seed()`. 
- Setting the seed makes your randomness **reproducible**: you will now get the same answers (in your knitted document) as your peers, provided you use the same code.

```r
set.seed(8362)
sample(1:5000, size = 3)
```

```
[1]  258 1834 2371
```

---
# Line Graphs 📈

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Remember this?

```r
sixSidedDieRoll <- function(n) {
  mean(sample(1:6, size = n, replace = T))
}
plot(sapply(1:300, sixSidedDieRoll), 
     main = "Law of Large Numbers Example",
     xlab = "Number of Six-Sided Dice",
     ylab = "Average")
```
]
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![](lab06-slides-async_files/figure-html/plotOfMeanOfEx1-1.png)
]

---
# Line Graphs 📈

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We can make a line graph with the `type` argument to `plot()`:

```r
plot(sapply(1:300, sixSidedDieRoll), 
     main = "Law of Large Numbers Example",
     xlab = "Number of Six-Sided Dice",
     ylab = "Average",
*    type = "l")
```

Use `type = l` for a `l`ine graph (that's a **lowercase L**)
]
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![](lab06-slides-async_files/figure-html/lineGraphL-1.png)
]

---
# Line Graphs 📈

.pull-left[

```r
plot(sapply(1:300, sixSidedDieRoll), 
     main = "Law of Large Numbers Example",
     xlab = "Number of Six-Sided Dice",
     ylab = "Average",
*    type = "o",
     pch = 20)
```

Use `type = o` to draw lines between points (and `pch` is back!)
]
.pull-right[
![](lab06-slides-async_files/figure-html/lineGraphO-1.png)
]

---

# Line Graphs 📈

.pull-left[

```r
plot(sapply(1:300, sixSidedDieRoll), 
     main = "Law of Large Numbers Example",
     xlab = "Number of Six-Sided Dice",
     ylab = "Average",
     type = "o",
     pch = 20,
*    lty = "dotted",
*    lwd = 2)
```

- Use `lty` to specify `l`ine `ty`pe: (0=blank, 1=solid (default), 2=dashed, 3=dotted, 4=dotdash, 5=longdash, 6=twodash)
- Use `lwd` to specify `l`ine `w`i`d`th (default is 1)
]
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![](lab06-slides-async_files/figure-html/lineGraphCustom-1.png)
]

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# Lab Project ⌨️

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### Your tasks
- Complete the "Try It!" and "Dive Deeper" portions of the lab assignment by copy/pasting and modifying appropriate code from earlier in the document.
]

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### How to get help
- Use the "labs" section of Piazza to ask questions and work with your peers.
- If you use Piazza, please note that in the "Collaborators" list at the top of the discussion section.
- If you're really stuck, email your lab instructor!
]

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