---
title: "Homework 3"
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date:
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html_document:
toc: true
toc_depth: 3
theme: paper
highlight: tango
---

##### This homework is due by **Sunday, midnight**. To complete this assignment, follow these steps:

1. Download the `homework3.Rmd` file from Blackboard or the course website.

2. Open `homework3.Rmd` in RStudio.

3. Replace the "Your Name Here" text in the `author:` field with your own name.

4. Supply your solutions to the homework by editing `homework3.Rmd`.

5. When you have completed the homework and have **checked** that your code both runs in the Console and knits correctly when you click `Knit HTML`, rename the R Markdown file to `homework3_YourNameHere.Rmd`, and submit on Blackboard. (YourNameHere should be changed to your own name.)

##### Homework tips:

1. Recall the following useful RStudio hotkeys.

|Keystroke    | Description |
|----------------|-----------------------------------------------------------------------------|
| `<tab>`    | Autocompletes commands and filenames, and lists arguments for functions.|
| `<up>`       | Cycles through previous commands in the console prompt |
| `<ctrl-up>` | Lists history of previous commands matching an unfinished one |
| `<ctrl-enter>` | Runs current line from source window to Console. Good for trying things out ideas from a source file. |
| `<ESC>`    | Aborts an unfinished command and get out of the + prompt |

**Note**: Shown above are the Windows/Linux keys. For Mac OS X, the `<ctrl>` key should be substituted with the `<command>` (âŒ˜) key.

2. Instead of sending code line-by-line with `<ctrl-enter>`, you can send entire code chunks, and even run all of the code chunks in your .Rmd file. Look under the <Chunks> menu of the Source panel.

3. Run your code in the Console and Knit HTML frequently to check for errors.

4. You may find it easier to solve a problem by interacting only with the Console at first, or by creating a separate `.R` source file that contains only R code and no Markdown.

### Problem 1: tapply() and aggregate()

This problem uses the birthwt dataset, which we load and manipulate in the code chunk below.

```{r}
library(MASS) # Needed for birthwt data
library(plyr) # Needed for mapvalues function

# Rename the columns to have more descriptive names
colnames(birthwt) <- c("birthwt.below.2500", "mother.age", "mother.weight",
"race", "mother.smokes", "previous.prem.labor", "hypertension", "uterine.irr",
"physician.visits", "birthwt.grams")

# Transform variables to factors with descriptive levels
birthwt <- transform(birthwt,
         race = as.factor(mapvalues(race, c(1, 2, 3),
                           c("white","black", "other"))),
         mother.smokes = as.factor(mapvalues(mother.smokes,
                           c(0,1), c("no", "yes"))),
         hypertension = as.factor(mapvalues(hypertension,
                           c(0,1), c("no", "yes"))),
         uterine.irr = as.factor(mapvalues(uterine.irr,
                           c(0,1), c("no", "yes")))
         )
```

##### (a) tapply() (two factors)

Use `with()` along with the `tapply()` function to produce a table showing the % of babies born weighing under 2500g within each combination of mother's race and smoking status.

```{r}
with(birthwt, tapply(birthwt.below.2500, INDEX = list(race, mother.smokes), FUN = function(x){100*sum(x)/length(x)}))
```

*Clarifying example*: Suppose that there were 35 babies born to white mothers who did not smoke, and that 7 of those babies had low birthweight. The (white, non-smoker) cell of the table would be equal to 7/35 = 20%. You *do not* need to display the % symbol in your table.

##### (b) tapply (three factors)

Use the `tapply()` function to produce a table showing the % of babies born weighing under 2500g, broken down by race, smoking status, and hypertension.

```{r}
with(birthwt, tapply(birthwt.below.2500,
                  INDEX = list(race, mother.smokes, hypertension),
                  FUN = function(x){100*sum(x)/length(x)}))
```

##### (c) aggregate() (three factors)

Repeat part (b) using the `aggregate()` function.

```{r}
res = aggregate(birthwt$birthwt.below.2500,
                  by = list(race = birthwt$race, mother.smokes = birthwt$mother.smokes, hypertension = birthwt$hypertension),
                  FUN = function(x){100*sum(x)/length(x)})
res
```

Simulation Problem 1: Rejection Sampling Consider the function =...