Bootstrap + Central Limit Theorm

Lecture 19

Dr. Elijah Meyer

Duke University
STA 199 - Spring 2023

March 24th, 2023

Checklist

– Clone ae-18

Announcements

– HW 4 Due Friday

– Lab 7 Due next Tuesday

– Project proposal feedback by today

– Peer Review Survey

– HW 6 - Statistical Experience

Warm Up

Our goal is to create a 95% confidence interval to estimate the true mean fipper length of penguins.

– What are the simulation methods called to create a confidence interval?

– How is one observation (“dot”) created when using such simulation methods?

Note: There are 344 penguins in the data set.

Warm Up

– What are the simulation methods called to create a confidence interval?

Bootstrap methods

Warm Up

– How is one observation (“dot”) created when using such simulation methods?

Sample WITH REPLACEMENT 344 (n) times
Calculate the new sample mean

Last Time

– Calculated bootstrap confidence intervals

– Interpreted confidence intervals

Last Time

– What is the difference between confidence and probability?

– Why is using the term probability incorrect when interpreting a confidence interval?

Confidence vs Probability

– Probability is simply how likely something is to happen

— This implies that the event has not happened yet!

— This implies that there is a random outcome associated with this event.

Probability coin example

What is the probability that this coin will land on heads?

Before the flip, what’s the probability that this coin lands on heads
What about when it lands?
What about if I cover it with my hand so you can’t see it?
What about if I flip it and cover it with my hand so you can’t see it?
Takeaway: Being unknown is not the same as being random

Confidence

Confidence - the percentage of all possible confidence intervals, created under the same conditions, expected to include the true population parameter

– For your confidence interval, the parameter is not random

Takeaway: Being unknown is not the same as being random

What you need to know

– Correct: We are 95% confident the true mean price per night for Airbnbs in Asheville is between 63.3 and 91.0 dollars.

– Incorrect: There is a 95% probability the true mean price per night for an Airbnb in Asheville is between 63.3 and 91.0.

For Today

– Central Limit Theorem (CLT)

– What impacts confidence intervals

– How to use CLT in practice

Goal of Statistical Inference

– Before we define CLT, let’s remind ourselves of our goal…

We are interested in population parameters, which we do not observe. So we calculate statistics from our sample to learn about them

As part of this process, we must quantify the degree of uncertainty in our sample statistic - which is why we make distributions!

Central Limit Theorem

– For a large enough n, the shape of the sampling distribution of the means is ~ normally distributed

– As n -> \(\infty\), the mean of the sample distribution is identical to the population mean \(\mu\) (so for a large n, our best guess of \(\mu\) is the mean of the sampling distribution)

– The standard deviation of the distribution of the sample means is \(\frac{s}{\sqrt(n)}\) (so for a large n, our best guess of \(\sigma\) is the standard deviation of the sampling distribution)

We will use the above guesses when we use a normal distribution to calculate a confidence interval

Assumptions

– Independence

– Sample Size

Independence

– Independent observations

– “Does one observation influence the other?”

Sample Size

– n > 30 (for quantitative variables)

– 10 successes + 10 failures (for categorical variables)

ae-18

We will walk through the CLT in action

Practice creating confidence intervals using the CLT and Bootstrap methods

Discussion

– Bootstrapping is less restrictive when it comes to sample size ( ~ n > 10)

– CLT does not work for other statistics besides the mean

– Both require the independence assumption