Getting Started

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 one-bedroom apartments in Manhattan.

Q - State the research question and identify the population and our sample.

  • Question:
  • Population:
  • Sample:

We start with loading relevant packages. For bootstrapping, we will use the infer package, included as part of tidymodels.

library(tidyverse)
library(tidymodels)
manhattan <- read_csv("data/manhattan.csv")

Part 1: Parameter and point estimate

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

  • Parameter: Mean rent of all one-bedroom apartments in Manhattan
  • Statistic: Mean rent of all apartments in the sample

Q - What is the point estimate of the mean rent?

Part 2: Manual bootstrapping

Let’s bootstrap! We should start by setting a seed. Function set.seed() is a base R function that allows us to control R’s random number generation. This ensures our analysis is reproducible, which means we’ll get the same random sample each time we run the code or knit.

Step 1: Sample with replacement.

Q - How many observations do we need for our bootstrap sample?

set.seed(123)

mht_boot1 <- manhattan %>% 
  slice_sample(n = ___, replace = ___)

# compare the original sample to the bootstrap sample
data.frame(org = manhattan$rent, 
           boot1 = mht_boot1$rent)

Step 2: Compute the statistic from the bootstrap sample.

Step 3: Repeat steps 1 and 2 multiple times and create a bootstrap distribution of sample statistics.

set.seed(123)

# how many times? 
reps <- 1000

# 1000 bootstrap samples 
boot_samp <- manhattan %>% 
  rep_slice_sample(n = 20, replace = TRUE, reps = reps)

# calculate sample mean from each bootstrap sample
boot_dist <- boot_samp %>% 
  group_by(replicate) %>% 
  summarize(stat = mean(rent))

# bootstrap distribution of sample means 
boot_dist %>% 
  ggplot(aes(x = stat)) + 
  geom_histogram(binwidth = 50) + 
  geom_vline(xintercept = mean(manhattan$rent), 
             color = "red", linetype = "dashed") +
  labs(x = "Sample mean", y = "Count", 
       title = "Bootstrap distribution of sample means", 
       subtitle = "Rent of one-bedroom apartments in Manhattan")

Step 4: Calculate a confidence interval (CI) using percentiles of the bootstrap distribution.

Q - What do you notice about the relationship between the confidence level and the width of the confidence interval?

Part 3: Bootstrap samples using functions in infer

Steps 1-3 can be done in one pipeline using functions in infer.

Q - Complete the code chunk below. Use 1000 reps for the in-class activity. (You will use about 15,000 reps for assignments outside of class.)

set.seed(123)

# save resulting bootstrap distribution
boot_dist2 <- manhattan %>%
  # specify the variable of interest
  specify(response = ____) %>%
  # generate reps (say 100, 1000, 10000, etc.) bootstrap samples
  generate(reps = ____, type = _____) %>%
  # calculate the statistic of each bootstrap sample
  calculate(stat = _____)
  • specify: what variable will you use?
  • generate: how many repetitions (samples) to create for the variable you specified? What type? type = draw, type = permute or type = bootstrap.
  • calculate: what statistic should be computed for each sample generated?

Order of functions matters! You can read more about each function using the help command, e.g., ?specify.

Q - How many rows are in boot_dist2?

Q - What does each row represent?

Q - What are the variables in boot_dist2? What do they mean?

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

Part 4: Confidence intervals

Q - Step 4. Construct the 95% confidence interval for the mean rent of one-bedroom apartments in Manhattan. To get middle 95%, we want to omit 2.5% on the left and on the right.

boot_dist2 %>% 
  summarize(lower_bound = quantile(____, ____), 
            upper_bound = quantile(____, 0.975))

Q - What is the correct interpretation for the interval calculated above?

Q - 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. Hint: upper - lower should yield what you want, e.g., 0.975 - 0.025 = 0.95.

Q - 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 90% or the 95% confidence interval?

Q - 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? Explain in terms of accuracy and precision.

Practice

Use bootstrapping to estimate the median rent for one-bedroom apartments in Manhattan.

  1. Generate the bootstrap distribution of the sample medians. Use 1000 reps. Save the results in boot_dist_median.
set.seed(123)

  1. Calculate a 92% confidence interval.

  2. Interpret the 92% confidence interval.

Submitting Application Exercises

  • Once you have completed the activity, push your final changes to your GitHub repo.
  • Make sure you committed at least three times.
  • Check that your repo is updated on GitHub, and that’s all you need to do to submit application exercises for participation.