---
title: "SDM4 in R: Sampling Distribution Models (Chapter 17)"
author: "Nicholas Horton (nhorton@amherst.edu) and Sarah McDonald"
date: "June 13, 2018"
output: 
  pdf_document:
    fig_height: 2.8
    fig_width: 6
  html_document:
    fig_height: 3
    fig_width: 5
  word_document:
    fig_height: 4
    fig_width: 6
---


```{r, include = FALSE}
# Don't delete this chunk if you are using the mosaic package
# This loads the mosaic and dplyr packages
require(mosaic)
```

```{r, include = FALSE}
# knitr settings to control how R chunks work.
require(knitr)
opts_chunk$set(
  tidy = FALSE,     # display code as typed
  size = "small"    # slightly smaller font for code
)
```

## Introduction and background 

This document is intended to help describe how to undertake analyses introduced 
as examples in the Fourth Edition of *Stats: Data and Models* (2014) by De Veaux, Velleman, and Bock.
More information about the book can be found at http://wps.aw.com/aw_deveaux_stats_series.  This
file as well as the associated R Markdown reproducible analysis source file used to create it can be found at http://nhorton.people.amherst.edu/sdm4.

This work leverages initiatives undertaken by Project MOSAIC (http://www.mosaic-web.org), an NSF-funded effort to improve the teaching of statistics, calculus, science and computing in the undergraduate curriculum. In particular, we utilize the `mosaic` package, which was written to simplify the use of R for introductory statistics courses. A short summary of the R needed to teach introductory statistics can be found in the mosaic package vignettes (http://cran.r-project.org/web/packages/mosaic).
A paper describing the mosaic approach was published in the *R Journal*: https://journal.r-project.org/archive/2017/RJ-2017-024.

## Chapter 17: Sampling Distribution Models

### Section 17.1: Sampling distribution of a proportion

Let's regenerate Figure 17.1 (page 444).

```{r}
library(mosaic)
options(digits = 3)
numsim <- 2000
n <- 1022
p <- 0.57
samples <- rbinom(numsim, size = n, prob = p)/n
gf_histogram(~ samples, xlab = "Distribution of sample proportions", 
          binwidth = 0.01, center = 0.01/2, type = "count")
gf_dist("norm", params = list(p, sqrt(p*(1-p)/n)), xlab = "p", ylab = "f(p)")
```

### Section 17.2: When does the normal model work?

We can replicate the example from page 449:

```{r}
p <- 0.22
n <- 200
pnorm(.155, mean = p, sd = sqrt(p*(1-p)/n))   # normal approximation
pbinom(31, size = n, prob = p)    # exact value
```


### Section 17.3: The sampling distribution of other statistics

Let's replicate the display on page 451:

```{r}
BodyFat <- read.csv("http://nhorton.people.amherst.edu/sdm4/data/Body_fat_complete.csv")
medians <- do(2000)*median(~ Weight, data = sample(BodyFat, 10, replace = FALSE))
gf_histogram(~ median, xlab = "Sampling distribution of medians", 
          binwidth = 5, center = 5/2, data = medians)
variances <- do(2000)*var(~ Weight, data = sample(BodyFat, 10, replace = FALSE))
gf_histogram(~ var, xlab = "Sampling distribution of variances", 
          binwidth = 100, center = 100/2, data = variances)
minimums <- do(2000)*min(~ Weight, data = sample(BodyFat, 10, replace = FALSE))
gf_histogram(~ min, xlab = "Sampling distribution of minimums", 
          binwidth = 2, center = 2/2, data = minimums)
```

Neither of the sampling distributions of the variance or the minimums are normally distributed.

### Section 17.4: Central Limit Theorem

Let's replicate the displays on pages 453-454:

```{r message=FALSE}
require(readr)
CEO <- read_delim("http://nhorton.people.amherst.edu/sdm4/data/CEO_Salary_2012.txt", 
                  delim = "\t")
CEO <- mutate(CEO, Pay = One_Year_Pay*1000)
favstats(~ Pay, data = CEO)
```

Note that Figure 17.11 seems to be off by a factor of 10!
```{r}
gf_histogram(~ Pay, xlab ="CEO Compensation in $1000", 
             binwidth = 10000, center = 10000/2-.01, data = CEO)
```


```{r}
samp10 <- do(2000)*mean(~ Pay, data = sample(CEO, 10))
gf_histogram(~ mean, xlab = "means of CEO Compensation (samples of size n = 10)", 
  binwidth = 2000, center = 2000/2, data = samp10)
```


```{r}
samp50 <- do(2000)*mean(~ Pay, data = sample(CEO, 50))
gf_histogram(~ mean, xlab = "means of CEO Compensation (samples of size n = 50)", 
  binwidth = 1000, center = 1000/2, data = samp50)
```

```{r}
samp100 <- do(2000)*mean(~ Pay, data = sample(CEO, 100))
gf_histogram(~ mean, xlab = "means of CEO Compensation (samples of size n = 100)", 
  binwidth = 500, center = 500/2, data = samp100)
```

```{r}
samp200<- do(2000)*mean(~ Pay, data = sample(CEO, 200))
gf_histogram(~ mean, xlab = "means of CEO Compensation (samples of size n = 200)", 
  binwidth = 500, center = 500/2, data = samp200)
```

Note how the axis limits get narrower as the sample size increases (since the means are less variable when the sample size increases:

```{r}
mysd <- sd(~ Pay, data = CEO) 
mysd
sd(~ mean, data = samp10)  # what we observed
mysd/sqrt(10)   # what we would expect
```

We can repeat this comparison for each of the sets of samples.

```{r}
sd(~ mean, data = samp50)
sd(~ mean, data = samp100)
sd(~ mean, data = samp200)
```