Questions about the Assignment
Understanding Inference:
Confidence Intervals II
Part I
The z-score is not the same as the percentile.
(e.g., a z-score of .98 does not equal the 98th percentile)
Confidence intervals
The z-score is the number of standard deviations the value is
above or below the mean.
Other levels of confidence
Bootstrap distribution
Part II
Some variables appear to be quantitative, but they are really
categorical (e.g. year born, income bracket, occupation code).
Good quantitative variables include phrases such as how many,
how often, etc. and the respondent answers with a number.
Who was asked the question (everyone or a subset of the sample).
Summary
The Problem
(From Last Class)
To create a plausible range of values for a population parameter:
1. Take many random samples from the population, and compute
the sample statistic for each sample.
2. Compute the standard error (i.e., the standard deviation of all
these statistics).
We have one random sample of the population.
We need multiple (i.e. >1,000) random samples from the
population to calculate the standard error of the sample statistic.
We cannot afford to conduct 1,000+ random samples.
3. The plausible range of values = sample statistic ± 2 × standard error
Is there a way to use the information from the one random sample
we have, to simulate 1,000+ random samples?
One small problem…
Often we only have one sample!
How can we calculate the variation in sample statistics,
if we only have one sample?
Random Sample
Population
(if the sample size is sufficiently large)
Yes, the Bootstrap simulation method enables us to generate a
sampling distribution.
Random Sample
Population
(if the sample size is sufficiently large)
When we conducted our
study yesterday, we
treated this bag of
Reese’s Pieces as the
population and we drew
random samples of size
10 from this population.
Is the proportion
of orange pieces in
this bag pretty
close to the
proportion of the
entire population
of Reese’s Pieces?
If our sample size was
500 (the total number of
pieces in the bag) we
would know the exact
proportion of the orange
pieces in the population.
Because this bag
is a random
sample of the
entire population
of all Reese’s
Pieces produced
by Hershey’s.
Why?
N = 500
n = 10
Hershey Factory
N = 1 Billion+
N = 500
1
Random Sample
Population
(if the sample size is sufficiently large)
Hershey Factory
N = 1 Billion+
p ≅.52
Population
(if the sample size is sufficiently large)
n = 500
̂ = .52
Random Sample
Population
Thus, we can draw
1,000+ samples
(n=500) from this
“population” to
generate the
equivalent of
1,000+ random
samples from the
population.
Reaching into this bag to draw
more samples, it’s equivalent to
reaching into the vat of Reese’s
Pieces at the Hershey Factory.
To generate a sampling
distribution, we can take
repeated random samples from
this “population”.
Our goal is to determine the
population proportion of orange
pieces.
We can use this
sample to generate
multiple samples.
Let’s say that we’ve determined
that a sample size of 500 is
sufficiently large enough to
adequately estimate the
population proportion.
Because this bag
is a random
sample of the
entire population,
it contains
approximately the
same proportion of
orange Reese’s
Pieces as the
entire population.
This bag is the one random
sample (n=500) of the
population we have.
It gives us one sample statistic,
but we need multiple sample
statistics.
(if the sample size is sufficiently large)
Drawing additional samples
from this bag is equivalent to
drawing additional samples
from the vat of Reese’s Pieces
at the Hershey Factory?
Population
(if the sample size is sufficiently large)
Because this bag
is a random
sample of the
entire population,
it contains
approximately the
same proportion of
orange Reese’s
Pieces as the
entire population.
Random Sample
Random Sample
n = 500
Sampling with Replacement
n = 500
Random Sample
Population
(if the sample size is sufficiently large)
But won’t all of the subsequent
samples of size 500 we draw
from this bag produce the same
sample statistic as the original
sample?
Yes, unless we sample with
replacement.
n = 500
Why “bootstrap”?
After we sample a unit, we put it back into the “population”
such that each unit can be selected more than once.
By sampling with replacement, we can ensure that the
“population” in the bag retains the same proportion of orange
pieces that are in the true population.
This type of sampling process is known as bootstrapping.
“Pull yourself up by your bootstraps”
Lift yourself up into the air simply by pulling up on the laces
of your boots.
A metaphor for accomplishing an “impossible” task without
any outside help.
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Bootstrapping Terms
Bootstrap sample: A random sample taken with replacement from
the original sample. It needs to be the same size as the original
sample.
Bootstrap sample statistic: The sample statistic computed on the
bootstrap sample.
Bootstrap sampling distribution: The sampling distribution of
many bootstrap sample statistics.
Original
Sample
Sample
Statistic
StatKey
Bootstrap
Sample
Bootstrap
Sample
Statistic
Bootstrap
Sample
Bootstrap
Sample
Statistic
.
.
.
.
.
.
Bootstrap
Sample
Bootstrap
Sample
Statistic
Bootstrap
Sampling
Distribution
Standard Error
www.lock5stat.com\statkey
The variability of the bootstrap statistics is similar to the
variability of the sample statistics.
count = 260
n = 500
The standard error of our sample statistic can be estimated
using the standard deviation of the bootstrap sampling
distribution.
Reese’s Pieces
Bootstrap Distribution
Based on this sample, give a 95% confidence interval for the
true proportion of Reese’s Pieces that are orange.
A. (0.50, 0.54)
B. (0.48, 0.56)
C. (0.48, 0.52)
D. (0.46, 0.54)
You have a sample of size n = 500. You sample with
replacement 1000 times to get 1000 bootstrap samples.
What is the sample size of each bootstrap sample?
0.52  2 × 0.02
Standard Deviation = .02
Sample
Mean
= .52
A. 500
B. 1,000
Bootstrap samples are the same
size as the original sample.
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Bootstrap Distribution
Bootstrap Distribution
You have a sample of size n = 500. You sample with
replacement 1000 times to get 1000 bootstrap samples.
You have a sample of size n = 500. You sample with
replacement 1000 times to get 1000 bootstrap samples.
How many bootstrap sample statistics will you have?
How many dots will be in a dotplot of the bootstrap sampling
distribution?
A. 1
B. 500
C. 1,000
Each bootstrap sample yields one
bootstrap sample statistic.
A. 50
B. 1,000
C. 50,000
Atlanta Commutes
Each dot in the bootstrap sampling
distribution corresponds to one
bootstrap sample statistic.
Random Sample of 500 Commutes
What’s the mean commute time for workers in metropolitan Atlanta?
Dot Plot
CommuteAtlanta
The Original Sample
n = 500
̅ = 29.11 minutes
s = 20.72 minutes
20
40
60
80
100
120
140
160
180
Time
This dotplot is…
A. the sample distribution of commute times.
B. the sampling distribution of sample statistics.
Random Sample of 500 Commutes
Random Sample of 500 Commutes
Dot Plot
CommuteAtlanta
Dot Plot
CommuteAtlanta
The Original Sample
The Original Sample
n = 500
̅ = 29.11 minutes
s = 20.72 minutes
n = 500
̅ = 29.11 minutes
s = 20.72 minutes
Hint: CI = sample statistic ± 2 × standard error
20
40
60
80
100
120
140
160
180
Time
The confidence interval for the point estimate is…
29.11 ± 2×20.72 is the interval that contains 95% of the
A. 29.11 20.72
B. 29.11 2 20.72 commute times in the original sample. The variability of
the sample statistic ̅ is
C. cannot be determined with the data available not known.
20
40
60
80
100
120
140
160
180
Time
How can we determine the variability of the sample statistic so that we can
calculate the confidence interval for the population parameter ( )?
Generate a Bootstrap Distribution using The Original Sample
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Bootstrap Distribution
www.lock5stat.com/statkey/
The Beauty of Bootstrapping
We can use bootstrapping to assess the uncertainty
surrounding any sample statistic.
The 95% Confidence Interval
point estimate ± the margin of error
If we have sample data, we can use bootstrapping to estimate
a 95% confidence interval for population parameter.
sample statistic ± 2 × standard error
sample statistic ± 2 × sd of the bootstrap
sampling distribution
29.11 ± 2 × 0.915
The 95% confidence interval for the average commute time is
A. (28.2, 30.0)
B. (27.3, 30.9)
C. (26.6, 31.8)
Obama’s Approval Rating
Obama’s Approval Rating
http://www.gallup.com/poll/113980/Gallup-Daily-Obama-Job-Approval.aspx
http://www.gallup.com/poll/113980/Gallup-Daily-Obama-Job-Approval.aspx
Gallup surveyed 1,500 Americans between June 9th-11th 2012 and 49%
of these people approved of the job Barack Obama is doing as president.
Sample statistic: (sample proportion)
̂ = .49
Calculate a 95% CI for the sample proportion.
www.lock5stat.com/statkey
Count = 735
n = 1,500
Obama’s Approval Rating
www.lock5stat.com/statkey
CI = (.464, .516)
Count = 735
N = 1500
Middle 95% of the bootstrap statistics
0.464
CI = original sample proportion
± 2 ×=
standard
Count
735 error
= .49 ± 2 × 0.013
N = 1500
= .49 ± .026 (remember Gallup’s
margin of error was ± .03
= (.464, .516)
We are 95% confident that the
true percentage of all Americans
that approve of Obama’s job
performance is between 46.4%
and 51.6%
Two Methods for
Calculating a 95% CI
Count
= 735
The Standard Error Method
or The
Percentile Method
N = 1500
CI = sample statistic ± 2 × standard error = (.464, .516)
Middle 95% of the bootstrap statistics
We are 95% confident that the
true percentage of all Americans
that approve of Obama’s job
performance is between 46.4%
and 51.6%
0.516
0.464
0.516
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Other Levels of Confidence
What if we want to be more than 95% confident?
How might you produce a 99% confidence interval for the
point estimate?
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.
www.lock5stat.com/statkey
Level of Confidence
The Effects of Sample Size
Which is wider, a 90% confidence interval or a 95%
confidence interval?
A. 90% CI
B. 95% CI
Are these bootstrap distributions the same?
n = 1500
n = 100
A 95% interval captures the middle 95%,
which is a wider range than the middle 90%
SE = .05
0.39
0.49
0.59
SE = .013
0.464
0.49
0.516
A bootstrap using a sample size of 1500 generates a standard error that is
much smaller than the standard error generated from a sample size of 100.
The margin of error decreases from 0.200 to 0.052.
Assignment
Finding Sample Proportions from the GSS
Part I: Graded Problems
3.74, and 3.76(a,c,d, and e)
Part II:
Goto http://sda.berkeley.edu/cgi-bin/hsda?harcsda+gss10
Enter the variable name here
For the following 3 categorical variables calculate the confidence interval for the
point estimate (i.e., sample proportion) using:
1. The Standard Error Method (show your work)
2. The Percentile Method (print/email your screenshot from www.lock5stat.com/statkey )
GENDER calculate the confidence interval for the proportion who are female
DIVORCE calculate the confidence interval for the proportion who’ve been divorced
GUNLAW calculate the confidence interval for the proportion who favor gun laws
Check “Column”
Check “Confidence Intervals”
Check “Unweighted”
Click on “Run Table”
6
Finding Sample Proportions from the GSS
StatKey and the Percentile Method
On the StatKey home page,
click on “CI for Single Proportion”
to get to this page.
This is the sample proportion
This is the confidence interval
This is the number of respondents
who indicated being female
This is the total number of
respondents
Click on “Edit Data” and a
window will pop up to enter:
n (the sample size)
count (the # of respondents
who are in the category
you are interested in
e.g., female)
Click here to generate 1000
bootstrap samples.
Click on “Two-Tail” to get the
95% confidence interval.
These values
represent the
95% confidence
interval.
Summary
The standard error of a statistic is the standard deviation of
the sampling distribution, which can be estimated from a
bootstrap distribution.
Confidence intervals for population parameter estimates can
be calculated using the standard error or the percentiles of a
bootstrap distribution.
Confidence intervals can be calculated this way as long as the
bootstrap distribution is approximately symmetric and
continuous.
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