Course Outline

segmentGetting Started (Don't Skip This Part)

segmentStatistics and Data Science: A Modeling Approach

segmentPART I: EXPLORING VARIATION

segmentChapter 1  Welcome to Statistics: A Modeling Approach

segmentChapter 2  Understanding Data

segmentChapter 3  Examining Distributions

segmentChapter 4  Explaining Variation

segmentPART II: MODELING VARIATION

segmentChapter 5  A Simple Model

segmentChapter 6  Quantifying Error

segmentChapter 7  Adding an Explanatory Variable to the Model

segmentChapter 8  Digging Deeper into Group Models

segmentChapter 9  Models with a Quantitative Explanatory Variable

segmentPART III: EVALUATING MODELS

segmentChapter 10  The Logic of Inference

segmentChapter 11  Model Comparison with F

segmentChapter 12  Parameter Estimation and Confidence Intervals

12.4 Using Bootstrapping to Calculate the 95% Confidence Interval

segmentChapter 13  What You Have Learned

segmentFinishing Up (Don't Skip This Part!)

segmentResources
list High School / Advanced Statistics and Data Science I (ABC)
12.4 Using Bootstrapping to Calculate the 95% Confidence Interval
Sliding a visualization up and down is a good way to understand the concept behind confidence intervals, but it’s not a very good way to calculate the actual upper and lower bounds! In this section we will learn one method (among many) to calculate a confidence interval.
In sliding the sampling distribution around, we’re making a few assumptions. We assume, first, that the shape and spread of the sampling distribution does not change as we slide it up and down the measurement scale. The sampling distribution is roughly normal for \(b_1\), which means it is unimodal and symmetrical, with a tail on either end.
We are also going to assume that the center of the confidence interval is at the observed sample \(b_1\) (e.g., 6.05 in the tipping study). We can convince you of this, we hope, using the picture below. We have recolored the sampling distribution centered at 6.05 in red. We also drew in two dotted black lines indicating the cutoffs that separate the likely and unlikely area of that sampling distribution. Behind it are the distributions we used to find the upper and lower bounds.
The .025 cutoff in the lower tail of the sampling distribution centered at the sample \(b_1\) lines up perfectly with the lower bound of the confidence interval. Likewise, the .025 cutoff in the upper tail lines up with the upper bound of the confidence interval. If only we had a sampling distribution centered at \(b_1\), we would be able to figure out the lower and upper bounds.
Bootstrapping with resample()
For calculating the confidence interval, it would be helpful to have a sampling distribution centered at the sample \(b_1\). Unfortunately, the shuffle()
function, which mimics a DGP where \(\beta_1=0\), produces a sampling distribution that is centered at 0. But we need to mimic a DGP where the \(\beta_1\) is equal to our sample \(b_1\) (6.05).
We can do this using the resample()
function. The resample()
function assumes that the entire population of interest is made up of observations that look just like the ones in our sample. In the case of the tipping experiment, we would assume a population made up of many copies of the tables in the TipExperiment
sample.
By repeatedly sampling from this imaginary population, we can create a sampling distribution of \(b_1\)s that will be centered at the observed sample \(b_1\). This approach to creating a sampling distribution is sometimes called bootstrapping.
We previously used the resample()
function with a vector (just a list of numbers) to simulate dice rolls. In bootstrapping, we will resample cases from a data frame instead of numbers from a vector.
To illustrate how this works, let’s focus on a subgroup of 6 tables from the TipExperiment
data frame. We have put these six tables in a new data frame called SixTables
. The output below shows the contents of this data frame.
TableID Tip Condition
4 34 Control
18 21 Control
43 21 Smiley Face
6 31 Control
25 47 Smiley Face
35 27 Smiley Face
Notice that in our small sample of 6 tables, there are 3 tables in the Smiley Face
condition and 3 in the Control
condition.
Now let’s see what happens when we resample()
from this sample of 6 tables.
resample(SixTables)
In the table below, we’ve put the original 6 tables on the left, and the results of the resample()
function on the right.
Original 6 Tables  Resampled 6 Tables 



The resample()
function takes a new random sample of six tables from the data set. It samples with replacement, meaning that when R randomly samples a table, it then puts the table back so it could potentially be sampled again. This explains why a table in the original data might appear more than once, or perhaps not at all, in the resampled data.
Okay, enough already with just six tables!
Let’s now use resample()
to bootstrap a new sample of 44 tables from the tables in the tipping study. Later, we will repeat this process many times to bootstrap a sampling distribution of \(b_1\)s. Let’s start by thinking about what would happen if you ran the following line of code on the complete TipExperiment
data frame:
resample(TipExperiment)
Both the resampled and original data frames will have 44 tables. However, because some tables might be selected more than once in the resampled data frame, and others not at all, the number of tables in each condition won’t exactly match the numbers in the original data frame. (We aren’t going to worry about this for now.)
We can also see that the average Tip
for each condition will be different in the resampled data frame. This makes sense because the exact tables included are not the same in the two data frames.
Use the code window below to produce the \(b_1\) estimate for the Condition
model of Tip
in both the original and resampled data frames. Run the code a few times, and see what you notice.
require(coursekata)
# run this a few times
b1(Tip ~ Condition, data = TipExperiment)
b1(Tip ~ Condition, data = resample(TipExperiment))
ex() %>% check_error()
Each time you run the code you will get two \(b_1\)s printed out. The first is based on the original data frame, and will always be 6.05; we know this by now! But the second \(b_1\) will vary each time you run the code. This is because each time you run the code, R is calculating the mean difference in tip percentage between smiley face and control group in a new resampled version of the data frame.