---
title: "Sampling Distributions <br> (Try 2)"
author: "Evan L. Ray"
date: "October 30, 2017"
output: ioslides_presentation
---

<style>
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    columns: 2 200px;         /* number of columns and width in pixels*/
    -webkit-columns: 2 200px; /* chrome, safari */
    -moz-columns: 2 200px;    /* firefox */
  }
  .col3 {
    columns: 3 100px;
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</style>

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = FALSE, cache = TRUE)
require(ggplot2)
require(scales)
require(dplyr)
require(tidyr)
require(readr)
require(mosaic)
```

## Sampling Distribution of Sample Mean

* The **sampling distribution** is the distribution of values of the sample mean, across all different samples

```{r, echo = FALSE, message = FALSE, warning = FALSE}
babies <- read_csv("https://mhc-stat140-2017.github.io/data/misc/babies1998/babies_dec_1998.csv")
babies <- filter(babies, !is.na(gestation))

set.seed(1)
```

```{r, echo = FALSE, fig.height=1.25}
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies) +
  geom_vline(mapping = aes(xintercept = mean(babies$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Population") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height=1.25}
babies_sample <- babies %>% sample_n(size = 10)
sample_means <- data.frame(
  sample_mean = mean(babies_sample$gestation)
)
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies_sample) +
  geom_vline(mapping = aes(xintercept = mean(babies_sample$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Sample, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height = 1.25}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = 1, data = sample_means) +
  xlim(range(babies$gestation)) +
  xlab("Sample Means, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```


## Sampling Distribution of Sample Mean

* The **sampling distribution** is the distribution of values of the sample mean, across all different samples

```{r, echo = FALSE, fig.height=1.25}
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies) +
  geom_vline(mapping = aes(xintercept = mean(babies$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Population") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height=1.25}
babies_sample <- babies %>% sample_n(size = 10)
sample_means <- bind_rows(
  sample_means,
  data.frame(
    sample_mean = mean(babies_sample$gestation)
  )
)
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies_sample) +
  geom_vline(mapping = aes(xintercept = mean(babies_sample$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Sample, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height = 1.25}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = 1, data = sample_means) +
  xlim(range(babies$gestation)) +
  xlab("Sample Means, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```


## Sampling Distribution of Sample Mean

* The **sampling distribution** is the distribution of values of the sample mean, across all different samples

```{r, echo = FALSE, fig.height=1.25}
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies) +
  geom_vline(mapping = aes(xintercept = mean(babies$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Population") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height=1.25}
babies_sample <- babies %>% sample_n(size = 10)
sample_means <- bind_rows(
  sample_means,
  data.frame(
    sample_mean = mean(babies_sample$gestation)
  )
)
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies_sample) +
  geom_vline(mapping = aes(xintercept = mean(babies_sample$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Sample, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height = 1.25}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = 1, data = sample_means) +
  xlim(range(babies$gestation)) +
  xlab("Sample Means, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```


## Sampling Distribution of Sample Mean

* The **sampling distribution** is the distribution of values of the sample mean, across all different samples

```{r, echo = FALSE, fig.height=1.25}
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies) +
  geom_vline(mapping = aes(xintercept = mean(babies$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Population") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height=1.25}
babies_sample <- babies %>% sample_n(size = 10)
sample_means <- bind_rows(
  sample_means,
  data.frame(
    sample_mean = mean(babies_sample$gestation)
  )
)
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies_sample) +
  geom_vline(mapping = aes(xintercept = mean(babies_sample$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Sample, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height = 1.25}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = 1, data = sample_means) +
  xlim(range(babies$gestation)) +
  xlab("Sample Means, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```


## Sampling Distribution of Sample Mean

* The **sampling distribution** is the distribution of values of the sample mean, across all different samples

```{r, echo = FALSE, fig.height=1.25}
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies) +
  geom_vline(mapping = aes(xintercept = mean(babies$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Population") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height=1.25}
babies_sample <- babies %>% sample_n(size = 10)
sample_means <- bind_rows(
  sample_means,
  data.frame(
    sample_mean = mean(babies_sample$gestation)
  )
)
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies_sample) +
  geom_vline(mapping = aes(xintercept = mean(babies_sample$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Sample, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height = 1.25}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = 1, data = sample_means) +
  xlim(range(babies$gestation)) +
  xlab("Sample Means, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```


## Sampling Distribution of Sample Mean

* The **sampling distribution** is the distribution of values of the sample mean, across all different samples

```{r, echo = FALSE, fig.height=1.25}
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies) +
  geom_vline(mapping = aes(xintercept = mean(babies$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Population") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height=1.25}
babies_sample <- babies %>% sample_n(size = 10)
sample_means <- bind_rows(
  sample_means,
  data.frame(
    sample_mean = mean(babies_sample$gestation)
  )
)
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies_sample) +
  geom_vline(mapping = aes(xintercept = mean(babies_sample$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Sample, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height = 1.25}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = 1, data = sample_means) +
  xlim(range(babies$gestation)) +
  xlab("Sample Means, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```


## Sampling Distribution of Sample Mean

* The **sampling distribution** is the distribution of values of the sample mean, across all different samples

```{r, echo = FALSE, fig.height=1.25}
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies) +
  geom_vline(mapping = aes(xintercept = mean(babies$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Population") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height=1.25}
babies_sample <- babies %>% sample_n(size = 10)
sample_means <- bind_rows(
  sample_means,
  data.frame(
    sample_mean = mean(babies_sample$gestation)
  )
)
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies_sample) +
  geom_vline(mapping = aes(xintercept = mean(babies_sample$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Sample, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height = 1.25}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = 1, data = sample_means) +
  xlim(range(babies$gestation)) +
  xlab("Sample Means, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```


## Sampling Distribution of Sample Mean

* The **sampling distribution** is the distribution of values of the sample mean, across all different samples

```{r, echo = FALSE, fig.height=1.25}
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies) +
  geom_vline(mapping = aes(xintercept = mean(babies$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Population") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height=1.25}
babies_sample <- babies %>% sample_n(size = 10)
sample_means <- bind_rows(
  sample_means,
  data.frame(
    sample_mean = mean(babies_sample$gestation)
  )
)
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies_sample) +
  geom_vline(mapping = aes(xintercept = mean(babies_sample$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Sample, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height = 1.25}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = 1, data = sample_means) +
  xlim(range(babies$gestation)) +
  xlab("Sample Means, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```


## Sampling Distribution of Sample Mean

* The **sampling distribution** is the distribution of values of the sample mean, across all different samples

```{r, echo = FALSE, fig.height=1.25}
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies) +
  geom_vline(mapping = aes(xintercept = mean(babies$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Population") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height=1.25}
babies_sample <- babies %>% sample_n(size = 10)
sample_means <- bind_rows(
  sample_means,
  data.frame(
    sample_mean = mean(babies_sample$gestation)
  )
)
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies_sample) +
  geom_vline(mapping = aes(xintercept = mean(babies_sample$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Sample, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height = 1.25}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = 1, data = sample_means) +
  xlim(range(babies$gestation)) +
  xlab("Sample Means, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```


## Sampling Distribution of Sample Mean

* The **sampling distribution** is the distribution of values of the sample mean, across all different samples

```{r, echo = FALSE, fig.height=1.25}
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies) +
  geom_vline(mapping = aes(xintercept = mean(babies$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Population") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height=1.25}
babies_sample <- babies %>% sample_n(size = 10)
sample_means <- bind_rows(
  sample_means,
  data.frame(
    sample_mean = mean(babies_sample$gestation)
  )
)
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies_sample) +
  geom_vline(mapping = aes(xintercept = mean(babies_sample$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Sample, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height = 1.25}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = 1, data = sample_means) +
  xlim(range(babies$gestation)) +
  xlab("Sample Means, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```


## Sampling Distribution of Sample Mean

* The **sampling distribution** is the distribution of values of the sample mean, across all different samples

```{r, echo = FALSE, fig.height=1.25}
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies) +
  geom_vline(mapping = aes(xintercept = mean(babies$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Population") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height=1.25}
babies_sample <- babies %>% sample_n(size = 10)
sample_means <- bind_rows(
  sample_means,
  data.frame(
    sample_mean = mean(babies_sample$gestation)
  )
)
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies_sample) +
  geom_vline(mapping = aes(xintercept = mean(babies_sample$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Sample, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height = 1.25}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = 1, data = sample_means) +
  xlim(range(babies$gestation)) +
  xlab("Sample Means, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```


## Sampling Distribution of Sample Mean

* The **sampling distribution** is the distribution of values of the sample mean, across all different samples of a certain size $n$.

```{r, echo = FALSE, fig.height=1.25}
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies) +
  geom_vline(mapping = aes(xintercept = mean(babies$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Population") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height=1.25}
sample_means <- bind_rows(
  sample_means,
  {do(10000) * {
    babies_sample <- babies %>% sample_n(size = 10)
    data.frame(
      sample_mean = mean(babies_sample$gestation)
    )
  }} %>% select(sample_mean)
)
```

```{r, echo = FALSE, warning = FALSE, fig.height=1.25}
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), alpha = 0, binwidth = 1, data = babies_sample) +
  geom_text(mapping = aes(x = x, y = y, label = label), size = 8,
    data = data.frame(x = 32, y = 0.15, label = "( ...draw 10,000 more samples of size n... )")) +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Sample, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, warning = FALSE, fig.height = 1.25}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = .2, data = sample_means) +
  xlim(range(babies$gestation)) +
  xlab("Sample Means, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```


## Sampling Distribution Depends on $n$

* Always centered at population mean, but as $n$ increases:
    * standard deviation gets smaller
    * distribution looks more normal

```{r, echo = FALSE, fig.height=1.05}
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies) +
  geom_vline(mapping = aes(xintercept = mean(babies$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Population") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, warning = FALSE, fig.height = 1.05}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = .2, data = sample_means) +
  xlim(range(babies$gestation)) +
  xlab("Sample Means, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, fig.height=1.05}
sample_means_1000 <- bind_rows(
  sample_means,
  {do(10000) * {
    babies_sample <- babies %>% sample_n(size = 1000)
    data.frame(
      sample_mean = mean(babies_sample$gestation)
    )
  }} %>% select(sample_mean)
)
```

```{r, echo = FALSE, warning = FALSE, fig.height = 1.05}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = .2, data = sample_means_1000) +
  xlim(range(babies$gestation)) +
  xlab("Sample Means, n = 1000") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```


## Sample Mean: Central Limit Theorem

* $Y_1, Y_2, \ldots, Y_n$ are independent observations of a quantitative variable
* Population has mean $\mu$ and standard deviation $\sigma$
* Compute the sample mean: $\bar{Y} = \frac{1}{n}\sum_{i=1}^n Y_i$
* The sampling distribution of $\bar{Y}$:
    * has mean $\mu$
    * has standard deviation $\sigma/\sqrt{n}$
    * for large enough $n$, it is approximately normal
    * putting this together: the sampling distribution of $\bar{Y}$ is approximately Normal($\mu$, $\sigma/\sqrt{n}$) for large enough $n$.

## More on $n$

* The sample size required for the sample mean be normally distributed depends on the population distribution

<div class="col2">
```{r, echo = FALSE, fig.height=1.15, fig.width = 3.5}
ggplot() +
  geom_histogram(mapping = aes(x = gestation, y = ..density..), binwidth = 1, data = babies) +
  geom_vline(mapping = aes(xintercept = mean(babies$gestation)), color = 'red') +
  xlim(range(babies$gestation)) +
  xlab("Gestation Time (weeks) -- Population") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, warning = FALSE, fig.height = 1.15, fig.width = 3.5}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = .2, data = sample_means) +
  xlim(range(babies$gestation)) +
  xlab("Sample Means, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE}
sample_means_100 <- bind_rows(
  sample_means,
  {do(10000) * {
    babies_sample <- babies %>% sample_n(size = 30)
    data.frame(
      sample_mean = mean(babies_sample$gestation)
    )
  }} %>% select(sample_mean)
)
```

```{r, echo = FALSE, warning = FALSE, fig.height = 1.15, fig.width = 3.5}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = .2, data = sample_means_100) +
  xlim(range(babies$gestation)) +
  xlab("Sample Means, n = 30") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, message = FALSE, warning = FALSE}
ceo_salaries <- read_csv("https://mhc-stat140-2017.github.io/data/sdm4/CEO_Salary_2012.csv")
ceo_salaries <- ceo_salaries %>%
  mutate(pay = `1-Year Pay ($mil)`) %>%
  filter(pay > 0)
```

```{r, echo = FALSE}
sample_means_10 <- do(10000) * {
    ceo_salaries_sample <- ceo_salaries %>% sample_n(size = 10)
    data.frame(
      sample_mean = mean(ceo_salaries_sample$pay)
    )
  }
```

```{r, echo = FALSE}
sample_means_100 <- do(10000) * {
    ceo_salaries_sample <- ceo_salaries %>% sample_n(size = 30)
    data.frame(
      sample_mean = mean(ceo_salaries_sample$pay)
    )
  }

```

```{r, echo = FALSE, fig.height=1.15, fig.width = 3.5, warning = FALSE}
ggplot() +
  geom_histogram(mapping = aes(x = pay, y = ..density..), binwidth = 1, data = ceo_salaries) +
  geom_vline(mapping = aes(xintercept = mean(ceo_salaries$pay)), color = 'red') +
  xlim(range(ceo_salaries$pay)) +
  xlab("CEO Pay (Millions of $) -- Population") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, warning = FALSE, fig.height = 1.15, fig.width = 3.5}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = .2, data = sample_means_10) +
  xlim(range(ceo_salaries$pay)) +
  xlab("Sample Means, n = 10") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```

```{r, echo = FALSE, warning = FALSE, fig.height = 1.15, fig.width = 3.5}
ggplot() +
  geom_histogram(mapping = aes(x = sample_mean, y = ..density..), binwidth = .2, data = sample_means_100) +
  xlim(range(ceo_salaries$pay)) +
  xlab("Sample Means, n = 30") +
  theme(plot.margin = unit(x = c(0, 0, 0, 0), units = "cm"))
```
</div>

## Estimating the Success Probability

* Suppose we want to estimate the proportion $p$ of US households who own the home they live in.

* We take a sample of size $n$ and count the number of households in our sample who own their home:

$$X \sim \text{Binomial}(n, p)$$

* How can we estimate $p$ using $X$?



## Sampling distribution of $\hat{p}$

* We will estimate the probability of success using
$$\hat{p} = \frac{X}{n}$$

* Remember that we can write $X$ as a sum of independent Bernoulli Random Variables:
$X = X_1 + X_2 + \cdots + X_n$

* So $\hat{p} = \frac{X}{n} = \frac{1}{n} \sum_i X_i$ is a sample mean of independent Bernoulli random variables

* Since $\hat{p} = \frac{1}{n} \sum_i X_i$, the Central Limit Theorem tells us the approximate sampling distribution of $\hat{p}$, for large enough $n$.

* https://istats.shinyapps.io/SampDist_Prop/


## Sampling distribution of $\hat{p}$

* For a single Bernoulli random variable,
    * $E(X_i) = p$
    * $SD(X_i) = \sqrt{p(1 - p)}$

* The CLT says that for large enough $n$, the sampling distribution of $\hat{p}$ is approximately $$\hat{p} \sim \text{Normal}(p, \sqrt{p(1 - p)/n})$$

* For estimating a proportion/probability $p$, we say $n$ is large enough if the **success/failures** condition is satisfied:
    * $np \geq 10$ and $n(1 - p) \geq 10$