Confidence and intervals

• Throughout the year, we’ve seen that:
  • Means are used for describing the typical value in a sample or population, but we usually don’t know what they are, because we can’t see the entire population.
  • Means of samples can be used to estimate means of populations.
  • By including a margin of error with our estimate, we create an interval that increases our confidence that we’ve located the correct value of the population mean.
• Today, we’ll learn how we can calculate margins of error by using a method called the bootstrap.
  • Which comes from the phrase, Picking yourself up by your own bootstraps.

In this lab

• Load the built-in atus (American Time Use Survey) dataset, which is a survey of how a sample of Americans spent their day.

  • The United States has an estimated population of 336,302,171 (as of April 15, 2024 9:10 a.m. PDT). How many people were surveyed for this particular dataset?

  • Note: If you want to know the current US population or world population in real time, click on this link: https://www.census.gov/popclock/

• The statistical investigative question we wish to answer is:

What is the mean age of people older than 15 living in the United States?

• Why is it important that the ATUS is a random sample?
• Use our atus data to calculate an estimate for the average age of people older than 15 living in the U.S.

One bootstrap

• A bootstrapped sample is when we take a random sample() of our original data (atus) WITH replacement.
  • The size of the sample should be the same size as the original data.
• We can create a single bootstrapped sample for the mean in three steps:
  1. Sample the number of the rows to use in our bootstrap.
  2. slice those rows from our original data into our bootstrap data.
  3. Calculate the mean our our bootstrapped data.

Our first bootstrap

• Fill in the blanks to sample the row numbers we’ll use in our bootstrapped sample.
  • Be sure to re-read what a bootstrapped sample is from the previous slide to help you fill in the blanks.
  • Use set.seed(123) before taking the sample.

bs_rows <- ____(1:____, size = ____, replace = ____)

• Use the slice function to create a new dataset that includes each row from our sample.

bs_atus <- slice(atus, bs_rows)

Take a look

• Look at the values of bs_rows and bs_atus.
• Write a paragraph that explains to someone that’s not familiar with R how you created bs_rows and bs_atus. Be sure to include an explanation of what the values of bs_rows mean and how those values are used to create bs_atus. Also, be sure to explain what each argument of each function does.

One strap, two strap

• Calculate the mean of the age variable in your bootstrapped data, then use a different value of set.seed() to create your own, personal bootstrapped sample. Then calculate its mean.
• Compare this second bootstrapped sample with three other classmates and write a sentence about how similar or different the bootstrapped sample means were.

Many bootstraps

• To use bootstrapped samples to create confidence intervals, we need to create many bootstrapped samples.
  • Normally, the more bootstrapped samples we use, the better the confidence interval.
  • In this lab, we’ll do() 500 bootstrapped samples.
• To make do()-ing 500 bootstraps easier, we’ll code our 3-step bootstrap method into a function.
  • Open a new R Script (File -> New File -> R Script) to write your function into.

Bootstrap function

• Fill in the blank space below with the 3 steps needed to create a bootstrapped sample mean for our atus data.
  • Each step should be written on its own line between the curly braces.

bs_func <- function() {
    
    
    
}

• Highlight and Run the code you write.

Visualizing our bootstraps

• Once your function is created, fill in the blanks to create 500 bootstrapped sample means:

bs_means <- do(____) * bs_func()

• Create a histogram for your bootstrapped samples and describe the center, shape and spread of its distribution.
  • These bootstrapped estimates no longer estimate the average age of people in the U.S.
  • Instead, they estimate how much the estimate of the average age of people in the U.S. varies.
• In the next slide, we’ll look at how we can use these bootstrapped means to create 90% confidence intervals.

Bootstrapped confidence intervals

• To create a 90% confidence interval, we need to decide between which two ages the middle 90% of our bootstrapped estimates are contained.
• Using your histogram, fill in the statement below:

The lowest 5% of our estimates are below _______ years and the highest 5% of our estimates are above_______ years.

• Use the quantile() function to check your estimates.
• Based on your bootstrapped estimates, between which two ages are we 90% confident the actual mean age of people living in the U.S. is contained?

On your own

• Using your bootstrapped sample means, create a 95% confidence interval for the mean age of people living in the U.S.
• Why is the 95% confidence interval wider than the 90% interval?
• Write down how you would explain what a 95% confidence interval means to someone not taking Introduction to Data Science.