1. Confidence Intervals: Bootstrap Distribution
2/6/12
More on confidence intervals
Bootstrap distribution
Other levels of confidence
Section 3.2, 3.3
Professor Kari Lock Morgan
Duke University

2. FEMMES http://www.duke.edu/web/FEMMES/ Program for 4-6th grade girls
Females Excelling More in Math, Engineering, and Science
Program for 4-6th grade girls
Saturday, Feb 18th, morning
I’ll be leading a statistics activity. Any girls interested in helping out? Let me know after class or me.

3. Confidence Intervals
The National Football Conference (NFC) has won 25 super bowls, while the American Football Conference (AFC) has won 21. You want to estimate the proportion of super bowls won by the NFC.
Would a confidence interval be appropriate?
Yes No

4. Confidence Intervals
Based on Nielson Media Research, last night’s super bowl received a 47.8 rating and 71 share.
“The rating is the percentage of television households tuned to a broadcast, and the share is the percentage of homes watching among those with TVs on at the time”
Nielson’s numbers are based on “a cross-section of representative homes throughout the U.S”.
You want to estimate the true rating and share. Would a confidence interval be appropriate?
Yes No  

5. Confidence Intervals
If context were added, which of the following would be an appropriate interpretation for a 95% confidence interval:
“we are 95% sure the interval contains the parameter”
“there is a 95% chance the interval contains the parameter”
Both (a) and (b)
Neither (a) or (b)

6. Summary from Last Class
To create a confidence interval:
Take many random samples from the population, and compute the sample statistic for each sample
Compute the standard error as the standard deviation of all these statistics
Use statistic  2SE
One small problem…

7. BOOTSTRAP! Reality … WE ONLY HAVE ONE SAMPLE!!!!
How do we know how much sample statistics vary, if we only have one sample?!?
BOOTSTRAP!

8. ONE Reese’s Pieces Sample
Sample: 52/100 orange
Where might the “true” p be?

9. “Population”
Imagine the “population” is many, many copies of the original sample
(What do you have to assume?)

10. Reese’s Pieces “Population”
Sample repeatedly from this “population”

11. Sampling with Replacement
To simulate a sampling distribution, we can just take repeated random samples from this “population” made up of many copies of the sample
In practice, we do this by sampling with replacement from the sample we have (each unit can be selected more than once)

12. Reese’s Pieces
How would you take a bootstrap sample from your sample of Reese’s Pieces?

13. Reese’s Pieces “Population”

14. Bootstrap
A bootstrap sample is a random sample taken with replacement from the original sample, of the same size as the original sample
A bootstrap statistic is the statistic computed on the bootstrap sample
A bootstrap distribution is the distribution of many bootstrap statistics

15. Bootstrap Distribution
BootstrapSample
Bootstrap Statistic
BootstrapSample
Bootstrap Statistic
Original Sample
Bootstrap Distribution . .
Sample Statistic
BootstrapSample
Bootstrap Statistic

16. “Pull yourself up by your bootstraps”
Why “bootstrap”?
“Pull yourself up by your bootstraps”
Lift yourself in the air simply by pulling up on the laces of your boots
Metaphor for accomplishing an “impossible” task without any outside help

17. lock5stat.com/statkey/

18. Golden Rule of Bootstrapping
Bootstrap statistics are to the original sample statistic as
the original sample statistic is to the population parameter

19. Standard Error
The variability of the bootstrap statistics is similar to the variability of the sample statistics
The standard error of a statistic can be estimated using the standard deviation of the bootstrap distribution!

20. Reese’s Pieces
Based on this sample, give a 95% confidence interval for the true proportion of Reese’s Pieces that are orange.
(0.47, 0.57)
(0.42, 0.62)
(0.41, 0.51)
(0.36, 0.56)
I have no idea
0.52  2 × 0.05

21. Bootstrap Distribution
You have a sample of size n = 50. You sample with replacement 1000 times to get 1000 bootstrap samples.
What is the sample size of each bootstrap sample? 50
1000

22. Bootstrap Distribution
You have a sample of size n = 50. You sample with replacement 1000 times to get 1000 bootstrap samples.
How many bootstrap statistics will you have? 50
1000

23. Bootstrap Distribution
You have a sample of size n = 50. You sample with replacement 1000 times to get 1000 bootstrap samples.
How many dots will be in a dotplot of the bootstrap distribution? 50
1000

24. Atlanta Commutes
What’s the mean commute time for workers in metropolitan Atlanta?
Data: The American Housing Survey (AHS) collected data from Atlanta in 2004

25. Random Sample of 500 Commutes
Where might the “true” μ be?
WE CAN BOOTSTRAP TO FIND OUT!!!

26. Original Sample

27. “Population” = many copies of sample Sample from this “population”

28. Bootstrap Distribution
lock5stat.com/statkey/
95% confidence interval for the average commute time for Atlantans:
(a) (28.2, 30.0) (b) (27.3, 30.9) (c) 26.6, (d) No idea

29. The Magic of Bootstrapping
We can use bootstrapping to assess the uncertainty surrounding ANY sample statistic!
If we have sample data, we can use bootstrapping to create a 95% confidence interval for any parameter!
(well, almost…)

30. “Is there solid evidence of global warming?”
What percentage of Americans believe in global warming?
A survey on 2,251 randomly selected individuals conducted in October 2010 found that 1328 answered “Yes” to the question
“Is there solid evidence of global warming?”
Source: “Wide Partisan Divide Over Global Warming”, Pew Research Center, 10/27/10.

31. Global Warming www.lock5stat.com/statkey
We are 95% sure that the true percentage of all Americans that believe there is solid evidence of global warming is between 57% and 61%

32. “Is there solid evidence of global warming?”
Does belief in global warming differ by political party?
“Is there solid evidence of global warming?”
The sample proportion answering “yes” was 79% among Democrats and 38% among Republicans.
(exact numbers for each party not given, but assume n=1000 for each group)
Give a 95% CI for the difference in proportions.
Source: “Wide Partisan Divide Over Global Warming”, Pew Research Center, 10/27/10.

33. Global Warming www.lock5stat.com/statkey
We are 95% sure that the difference in the proportion of Democrats and Republicans who believe in global warming is between 0.37 and 0.45.

34. Global Warming
Based on the data just analyzed, can you conclude that the proportion of people believing in global warming differs by political party?
(a) Yes
(b) No

35. Body Temperature What is the average body temperature of humans?
We are 95% sure that the average body temperature for humans is between 98.05 and 98.47
98.6???
Shoemaker, What's Normal: Temperature, Gender and Heartrate, Journal of Statistics Education, Vol. 4, No. 2 (1996)

36. Other Levels of Confidence
What if we want to be more than 95% confident?
How might you produce a 99% confidence interval for the average body temperature?

37. (0.5th percentile, 99.5th percentile)
Percentile Method
For a P% confidence interval, keep the middle P% of bootstrap statistics
For a 99% confidence interval, keep the middle 99%, leaving 0.5% in each tail.
The 99% confidence interval would be
(0.5th percentile, 99.5th percentile)
where the percentiles refer to the bootstrap distribution.

38. Body Temperature www.lock5stat.com/statkey
We are 99% sure that the average body temperature is between 98.00 and 98.58

39. Level of Confidence
Which is wider, a 90% confidence interval or a 95% confidence interval?
(a) 90% CI
(b) 95% CI

40. Two Methods for 95%

41. Two Methods
For a symmetric, bell-shaped bootstrap distribution, using either the standard error method or the percentile method will given similar 95% confidence intervals
If the bootstrap distribution is not bell-shaped or if a level of confidence other than 95% is desired, use the percentile method

42. Mercury and pH in Lakes: Correlation
We are 90% confident that the true correlation between average mercury level and pH of Florida lakes is between and
Lange, Royals, and Connor, Transactions of the American Fisheries Society (1993)

43. Summary
The standard error of a statistic is the standard deviation of the sample statistic, which can be estimated from a bootstrap distribution
Confidence intervals can be created using the standard error or the percentiles of a bootstrap distribution
Confidence intervals can be created this way for any parameter, as long as the bootstrap distribution is approximately symmetric and continuous

44. To Do Homework 3 (due Monday) Research question and data for Project 1
(proposal due 2/15)
As soon as you come up with a question and find data, you already know how to do the first half of Project 1!
If you haven’t yet registered your clicker, me with the number on the back