Chapter 8 Bootstrapping and Confidence Intervals

In Chapter 7, we studied sampling. We started with a “tactile” exercise where we wanted to know the proportion of balls in the sampling bowl in Figure 7.1 that are red. While we could have performed an exhaustive count, this would have been a tedious process. So instead, we used a shovel to extract a sample of 50 balls and used the resulting proportion that were red as an estimate. Furthermore, we made sure to mix the bowl’s contents before every use of the shovel. Because of the randomness created by the mixing, different uses of the shovel yielded different proportions red and hence different estimates of the proportion of the bowl’s balls that are red.

We then mimicked this “tactile” sampling exercise with an equivalent “virtual” sampling exercise performed on the computer. Using our computer’s random number generator, we quickly mimicked the above sampling procedure a large number of times. In Subsection 7.2.4, we quickly repeated this sampling procedure 1000 times, using three different “virtual” shovels with 25, 50, and 100 slots. We visualized these three sets of 1000 estimates in Figure 7.15 and saw that as the sample size increased, the variation in the estimates decreased.

In doing so, what we did was construct sampling distributions. The motivation for taking 1000 repeated samples and visualizing the resulting estimates was to study how these estimates varied from one sample to another; in other words, we wanted to study the effect of sampling variation. We quantified the variation of these estimates using their standard deviation, which has a special name: the standard error. In particular, we saw that as the sample size increased from 25 to 50 to 100, the standard error decreased and thus the sampling distributions narrowed. Larger sample sizes led to more precise estimates that varied less around the center.

We then tied these sampling exercises to terminology and mathematical notation related to sampling in Subsection 7.3.1. Our study population was the large bowl with \(N\) = 2400 balls, while the population parameter, the unknown quantity of interest, was the population proportion \(p\) of the bowl’s balls that were red. Since performing a census would be expensive in terms of time and energy, we instead extracted a sample of size \(n\) = 50. The point estimate, also known as a sample statistic, used to estimate \(p\) was the sample proportion \(\widehat{p}\) of these 50 sampled balls that were red. Furthermore, since the sample was obtained at random, it can be considered as unbiased and representative of the population. Thus any results based on the sample could be generalized to the population. Therefore, the proportion of the shovel’s balls that were red was a “good guess” of the proportion of the bowl’s balls that are red. In other words, we used the sample to infer about the population.

However, as described in Section 7.2, both the tactile and virtual sampling exercises are not what one would do in real life; this was merely an activity used to study the effects of sampling variation. In a real-life situation, we would not take 1000 samples of size \(n\), but rather take a single representative sample that’s as large as possible. Additionally, we knew that the true proportion of the bowl’s balls that were red was 37.5%. In a real-life situation, we will not know what this value is. Because if we did, then why would we take a sample to estimate it?

An example of a realistic sampling situation would be a poll, like the Obama poll you saw in Section 7.4. Pollsters did not know the true proportion of all young Americans who supported President Obama in 2013, and thus they took a single sample of size \(n\) = 2089 young Americans to estimate this value.

So how does one quantify the effects of sampling variation when you only have a single sample to work with? You cannot directly study the effects of sampling variation when you only have one sample. One common method to study this is bootstrapping resampling, which will be the focus of the earlier sections of this chapter.

Furthermore, what if we would like not only a single estimate of the unknown population parameter, but also a range of highly plausible values? Going back to the Obama poll article, it stated that the pollsters’ estimate of the proportion of all young Americans who supported President Obama was 41%. But in addition it stated that the poll’s “margin of error was plus or minus 2.1 percentage points.” This “plausible range” was [41% - 2.1%, 41% + 2.1%] = [38.9%, 43.1%]. This range of plausible values is what’s known as a confidence interval, which will be the focus of the later sections of this chapter.

Needed packages

Let’s load all the packages needed for this chapter (this assumes you’ve already installed them). Recall from our discussion in Section 4.4 that loading the tidyverse package by running library(tidyverse) loads the following commonly used data science packages all at once:

  • ggplot2 for data visualization
  • dplyr for data wrangling
  • tidyr for converting data to tidy format
  • readr for importing spreadsheet data into R
  • As well as the more advanced purrr, tibble, stringr, and forcats packages

If needed, read Section 1.3 for information on how to install and load R packages.