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By the end of today you will…

Getting started

Download this application exercise by pasting the code below into your console

download.file("https://sta101.github.io/static/appex/ae9.Rmd",
destfile = "ae9.rmd")

Load packages

library(tidyverse)
library(tidymodels)
manhattan = read_csv("https://sta101.github.io/static/appex/data/manhattan.csv")
## Rows: 20 Columns: 1
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## dbl (1): rent
## 
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

Notes (for reference)

Bootstrapping is a re-sampling technique. The key idea is you have already collected a sample of size \(N\) from the population. To create a bootstrap sample, you sample with replacement from your original sample \(N\) times.

Let’s say you measure the height of five Duke students in meters:

heights = c(1.51, 1.62, 1.89, 2.01, 1.78)

students = data.frame(heights)

There are many ways to perform a bootstrap sample in R.

Method 1

set.seed(1)
sample(heights, size = 5, replace = TRUE)
## [1] 1.51 2.01 1.51 1.62 1.78

Method 2

set.seed(2)
students %>%
  specify(response = heights) %>%
  generate(reps = 1, type = "bootstrap")
## Response: heights (numeric)
## # A tibble: 5 × 2
## # Groups:   replicate [1]
##   replicate heights
##       <int>   <dbl>
## 1         1    1.78
## 2         1    1.51
## 3         1    1.78
## 4         1    1.51
## 5         1    2.01

From here, we can compute a bootstrap statistic. E.g.

set.seed(1)
sample(heights, size = 5, replace = TRUE) %>%
  median()
## [1] 1.62
set.seed(2)
students %>%
  specify(response = heights) %>%
  generate(reps = 1, type = "bootstrap") %>%
  calculate(stat = "median")
## Response: heights (numeric)
## # A tibble: 1 × 1
##    stat
##   <dbl>
## 1  1.78

  • take-away: sample() takes individual columns which can be handy for quick sampling while tidy syntax, specify() %>% generate() %>% calculate() plays nice with whole data frames and sets us up to easily implement future use-cases.

  • the tidy way uses the infer package (included as a part of tidymodels)

Example: rent in Manhattan

On a given day in 2018, twenty one-bedroom apartments were randomly selected on Craigslist Manhattan from apartments listed as “by owner”. The data are in the manhattan data frame. We will use this sample to conduct inference on the typical rent of 1 bedroom apartments in Manhattan.

Part 1: Drawing a bootstrap sample

Let’s start by using bootstrapping to estimate the mean rent of one-bedroom apartments in Manhattan.

Exercise 1

What is a point estimate (i.e. single number summary) of the typical rent?

Exercise 2

Let’s bootstrap!

  • To bootstrap we will sample with replacement by drawing a value from the box.
  • How many draws do we need for our bootstrap sample?

Fill in the values from the bootstrap sample conducted in class. Once the values are filled in, un-comment the code.

# class_bootstrap = c()

Exercise 3

  • About what value do you expect the bootstrap statistic to take?
  • Calculate the statistic from the bootstrap sample.
# add code

Part 2: Bootstrap confidence interval

We will use the infer package, included as part of tidymodels to calculate a 95% confidence interval for the mean rent of one-bedroom apartments in Manhattan.

We start by setting a seed to ensure our analysis is reproducible.

Generating the bootstrap distribution

We can use R to take many bootstrap samples, compute a statistic and then view the bootstrap distribution of that statistic.

Un-comment the lines and fill in the blanks to create the bootstrap distribution of sample means and save the results in the data frame boot_dist.

Use 1000 reps for the in-class activity. (You will use about 10,000 reps for assignments outside of class.)

set.seed(7182022)

boot_dist = manhattan #%>%
  #specify(______) %>%
  #generate(______) %>%
  #calculate(______)
  • How many rows are in boot_dist?
  • What does each row represent?
  • What are the variables in boot_dist? What do they mean?

Visualize the bootstrap distribution

A sample statistic is a random variable, we can look at its distribution.

Visualize the bootstrap distribution using a histogram. Describe the shape, center, and spread of this distribution.

# add code

Calculate the confidence interval

Uncomment the lines and fill in the blanks to construct the 95% bootstrap confidence interval for the mean rent of one-bedroom apartments in Manhattan.

#___ %>%
#  summarize(lower = quantile(______),
  #          upper = quantile(______))

Interpret the interval

Write the interpretation for the interval calculated above.

Part 3: Changing the confidence level

  • Modify the code used to calculate a 95% confidence interval to calculate a 90% confidence interval for the mean rent of one-bedroom apartments in Manhattan. How does the width of this interval compare to the width of the 95% confidence interval?
#calculate a 90% confidence interval
  • Now let’s calculate a 99% confidence interval for the mean rent of one-bedroom apartments in Manhattan. How does the width of this interval compare to the width of the 95% confidence interval?
#calculate a 99% confidence interval

  • Question: Does a confidence interval have to be symmetric?

  • What is one advantage to using a 90% confidence interval instead of a 95% confidence interval to estimate a parameter? - What is one advantage to using a 99% confidence interval instead of a 95% confidence interval to estimate a parameter?

Part 4: Additional practice

Next, use bootstrapping to estimate the median rent for one-bedroom apartments in Manhattan. - Generate the bootstrap distribution of the sample medians. Use 100 reps. Save the results in boot_dist_median.

set.seed(100)
## add code
  • Visualize the distribution. What do you notice?
# code here
  • Calculate a 92% confidence interval.
## add code
  • Interpret the 92% confidence interval.