Statistics and Data
Analysis
Professor William Greene
Stern School of Business
IOMS Department
Department of Economics
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Part 12: Statistical Inference
Statistics and Data Analysis
Part 12 – Statistical
Inference:
Confidence Intervals
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Part 12: Statistical Inference
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Part 12: Statistical Inference
Statistical Inference:
Point Estimates and
Confidence Intervals
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Statistical Inference
Estimation of Population Features Using Sample
Data
Sampling Distributions of Statisticss
Point Estimates and the Law of Large Numbers
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Uncertainty in Estimation
Interval Estimation
Part 12: Statistical Inference
Application: Credit Modeling
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1992 American Express analysis of
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Application process: Acceptance or rejection
Cardholder behavior
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Loan default
Average monthly expenditure
General credit usage/behavior
13,444 applications in November, 1992
Part 12: Statistical Inference
Modeling Fair
Isaacs’s Acceptance Rate
13,444 Applicants for a Credit Card (November, 1992)
Experiment = A
randomly picked
application.
Let X = 0 if Rejected
Let X = 1 if Accepted
Rejected
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Approved
Part 12: Statistical Inference
The Question They Are
Really Interested In: Default
Of 10,499 people whose application
was accepted, 996 (9.49%)
defaulted on their credit account
(loan). We let X denote the
behavior of a credit card recipient.
X = 0 if no default
(Bernoulli)
X = 1 if default
This is a crucial variable for a
lender. They spend endless
resources trying to learn more
about it. Mortgage providers in
2000-2007 could have, but
deliberately chose not to.
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Part 12: Statistical Inference
The data contained many covariates. Do these help
explain the interesting variable?
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Part 12: Statistical Inference
Variables Typically Used
By Credit Scorers
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Part 12: Statistical Inference
Sample Statistics
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The population has characteristics
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Mean, variance
Median
Percentiles
A random sample is a “slice” of the population
Part 12: Statistical Inference
Populations and Samples
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Population features of a
random variable.
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Mean = μ
= expected value of a
random variable
Standard deviation = σ
= (square root) of expected
squared deviation of the
random variable from the
mean
Percentiles such as the
median = value that divides
the population in half – a
value such that 50% of the
population is below this
value
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Sample statistics that
describe the data
 Sample mean = x
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= the average value in the
sample
Sample standard deviation = s
tells us where the sample
values will be (using our
empirical rule, for example)
Sample median helps to
locate the sample data on a
figure that displays the data,
such as a histogram.
Part 12: Statistical Inference
The Overriding Principle in
Statistical Inference
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The characteristics of a random sample will
mimic (resemble) those of the population
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Mean, median, standard deviation, etc.
Histogram
The resemblance becomes closer as the
number of observations in the (random)
sample becomes larger. (The law of large
numbers)
Part 12: Statistical Inference
Point Estimation
We use sample features to estimate
population characteristics.
 Mean of a sample from the population is
an estimate of the mean of the
population:
x is an estimator of μ
 The standard deviation of a sample from
the population is an estimator of the
standard deviation of the population:
s is an estimator of σ
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Part 12: Statistical Inference
Point Estimator
A formula
 Used with the sample data to estimate a
characteristic of the population (a
parameter)
 Provides a single value:
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Ni1xi
x
 a point estimator of 
N
 (xi  x)
s
 a point estimator of 
N 1
N
i1
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2
Part 12: Statistical Inference
Use random samples
and basic descriptive
statistics.
What is the ‘breach
rate’ in a pool of tens
of thousands of
mortgages? (‘Breach’
= improperly
underwritten or
serviced or otherwise
faulty mortgage.)
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Part 12: Statistical Inference
The forensic analysis was an examination of
statistics from a random sample of 1,500 loans.
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Part 12: Statistical Inference
Sampling Distribution
The random sample is itself random, since
each member is random.
 Statistics computed from random samples will
vary as well.
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Part 12: Statistical Inference
Estimating Fair
Isaacs’s Acceptance Rate
13,444 Applicants for a Credit Card (November, 1992)
Experiment = A
randomly picked
application.
Let X = 0 if Rejected
Let X = 1 if Accepted
Rejected
Approved
The 13,444 observations are the population. The true proportion is μ = 0.780943. We
draw samples of N from the 13,444 and use the observations to estimate μ.
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Part 12: Statistical Inference
The Estimator
The sample proportion we are examining here
is a sample mean.
X = 0 if the individual's application is rejected
X = 1 if the individual's application is accepted
1 N
The "acceptance rate" is x =  x i.
N i1
The population proportion is  = 0.780943.
x is an estimator of , the population mean.
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Part 12: Statistical Inference
x in 100 samples with N = 144 in each sample
0.780943 is the true proportion in the population we are sampling from.
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Part 12: Statistical Inference
The Mean is A Good Estimator
Sometimes x is too high, sometimes too low.
On average, it seems to be right.
The sample mean of the 100 sample estimates is 0.7844
The population mean (true proportion)
is 0.7809.
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Part 12: Statistical Inference
What Makes it a Good Estimator?
The average of the averages will hit
the true mean (on average)
 The mean is UNBIASED
(No moral connotations)
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Part 12: Statistical Inference
What Does the Law of
Large Numbers Say?
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The sampling variability in the estimator
gets smaller as N gets larger.
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If N gets large enough, we should hit the
target exactly; The mean is
CONSISTENT
Part 12: Statistical Inference
.7 to .88
N=144
.7 to .88
N=1024
.7 to .88
N=4900
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Part 12: Statistical Inference
Uncertainty in Estimation
How to quantify the variability in
the proportion estimator
--------+--------------------------------------------------------------------Variable|
Mean
Std.Dev.
Minimum
Maximum
Cases Missing
--------+--------------------------------------------------------------------Average of the means of the 100 samples of 144 observations
RATES144|
.78444
.03278
.715278
.868056
100
0
Average of the means of the 100 samples of 1024 observations
RATE1024|
.78366
.01293
.754883
.812500
100
0
Average of the means of the 100 samples of 4900 observations
RATE4900|
.78079
.00461
.770000
.792449
100
0
--------+---------------------------------------------------------------------
The population mean (true proportion)
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is 0.7809.
Part 12: Statistical Inference
Range of Uncertainty
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The point estimate will be off (high or low)
Quantify uncertainty in ± sampling error.
Look ahead: If I draw a sample of 100, what value(s) should I
expect?
 Based on unbiasedness, I should expect the mean to hit the
true value.
 Based on my empirical rule, the value should be within plus or
minus 2 standard deviations 95% of the time.
What should I use for the standard deviation?
Part 12: Statistical Inference
Estimating the Variance of the
Distribution of Means
We will have only one sample!
 Use what we know about the variance of
the mean:
 Var[mean] = σ2/N
N
2
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i1(xi  x)
Estimate
using the data: s 
N 1
Then, divide s2 by N.
σ2
2
Part 12: Statistical Inference
The Sampling Distribution
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For sampling from the population and
using the sample mean to estimate the
population mean:
 Expected value of x will equal μ
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Standard deviation of x
will equal σ/ √ N
CLT suggests a normal distribution
Part 12: Statistical Inference
The sample mean for a
given sample may be
very close to the true
mean
The sample mean for a
given sample may be
quite far from the true
mean
This is the sampling variability of the mean as an estimator of μ
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Part 12: Statistical Inference
Recognizing
Sampling Variability
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To describe the distribution of sample means, use the
sample x to estimate the population expected value
To describe the variability, use the sample standard
deviation, s, divided by the square root of N
To accommodate the distribution, use the empirical rule,
95%, 2 standard deviations.
Part 12: Statistical Inference
Estimating the Sampling Variability
For one of the samples, the mean was
0.849, s was 0.358. s/√N = .0298. If this
were my estimate, I would use
0.849 ± 2 x 0.0298
 For a different sample, the mean was
0.750, s was 0.433, s/√N = .0361. If this
were my estimate I would use
0.750 ± 2 x 0.0361
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Part 12: Statistical Inference
Estimates plus and minus two standard errors
The interval mean ± 2 standard errors almost always includes the true value of
.7809. The arrows show the cases in which the interval does not contain .7809.
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Part 12: Statistical Inference
How to use these results
The sample mean is my best guess of the
population mean.
 I must recognize that there will be estimation
error because of random sampling.
 I use the confidence interval to suggest a range
of plausible values for the mean, based on my
sample information.
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Part 12: Statistical Inference
Will the Interval
Contain the True Value?
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Uncertain: The midpoint is random; it may be
very high or low, in which case, no. Sometimes
it will contain the true value.
The degree of certainty depends on the width of
the interval.
 Very narrow interval: very uncertain.
(1 standard errors)
 Wide interval: much more certain
(2 standard errors)
 Extremely wide interval: nearly perfectly
certain (2.5 standard errors)
 Infinitely wide interval: Absolutely certain.
Part 12: Statistical Inference
The Degree of Certainty
The interval is a “Confidence Interval”
 The degree of certainty is the degree of
confidence.
 The standard in statistics is 95% certainty
(about two standard errors).
 I can be more confident if I make the
interval wider.
 I can be 100% confident if I make the
interval ‘infinitely’ wide. This is not helpful.
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Part 12: Statistical Inference
67 % and 95% Confidence Intervals
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Part 12: Statistical Inference
Monthly Spending Over First 12 Months
Population = 10,239
individuals who
(1) Received the Card
(2) Used the card at
least once
(3) Monthly
spending no more
than 2500.
What is the true mean
of the population that
produced these data?
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Part 12: Statistical Inference
Estimating the Mean
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Given a sample
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x
= 241.242
 S = 276.894
Estimate the population mean
 Point estimate 241.242
 66⅔% confidence interval: 241.242 ± 1 x 276.894/√225
= 227.78 to 259.70
 95% confidence interval: 241.242 ± 2 x 276.894/√225
= 204.32 to 278.162
 99% confidence interval: 241.242 ± 2.5 x 276.894/√225
= 195.09 to 287.39
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N = 225 observations
Part 12: Statistical Inference
Where Did the Interval Widths
Come From?
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Empirical rule of thumb:
 2/3 = 66 2/3% is contained in an interval that is the mean plus
and minus 1 standard deviation
 95% is contained in a 2 standard deviation interval
 99% is contained in a 2.5 standard deviation interval.
Based exactly on the normal distribution, the exact values would be
 0.9675 standard deviations for 2/3 (rather than 1.00)
 1.9600 standard deviations for 95% (rather than 2.00)
 2.5760 standard deviations for 99% (rather than 2.50)
Part 12: Statistical Inference
Large Samples
If the sample is moderately large (over
30), one can use the normal distribution
values instead of the empirical rule.
 The empirical rule is easier to remember.
The values will be very close to each
other.
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Part 12: Statistical Inference
Refinements (Important)
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When you have a fairly small sample (under 30)
and you have to estimate σ using s, then both
the empirical rule and the normal distribution
can be a bit misleading. The interval you are
using is a bit too narrow.
You will find the appropriate widths for your
interval in the “t table” The values depend on
the sample size. (More specifically, on
N-1 = the degrees of freedom.)
Part 12: Statistical Inference
Critical Values
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For 95% and 99% using a sample of 15:
 Normal:
1.960 and 2.576
 Empirical rule: 2.000 and 2.500
 T[14] table:
2.145 and 2.977
Note that the interval based on t is noticeably wider.
The values from “t” converge to the normal values
(from above) as N increases.
What should you do in practice? Unless the sample is
quite small, you can usually rely safely on the empirical
rule. If the sample is very small, use the t distribution.
Part 12: Statistical Inference
n = N-1
Small
sample
Large
sample
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Part 12: Statistical Inference
Application
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A sports training center is examining the endurance of athletes. A
sample of 17 observations on the number of hours for a specific task
produces the following sample:
4.86, 6.21, 5.29, 4.11, 6.19, 3.58, 4.38, 4.70, 4.66,
5.64, 3.77, 2.11, 4.81, 3.31, 6.27, 5.02, 6.12
This being a biological measurement, we are confident that the
underlying population is normal.
Form a 95% confidence interval for the mean of the distribution.
The sample mean is 4.766. The sample standard deviation, s, is 1.160.
The standard error of the mean is 1.16/√17 = 0.281.
Since this is a small sample from the normal distribution, we use the
critical value from the t distribution with N-1 = 16 degrees of freedom.
From the t table (previous page), the value of t[.025,16] is 2.120
The confidence interval is 4.766 ± 2.120(0.281) = [4.170,5.362]
Part 12: Statistical Inference
Application: The Margin of Error
The % is a mean of Bernoulli
variables, Xi = 1 if the respondent
favors the candidate, 0 if not.
The % equals 100[(1/652)Σixi].
(1) Why do they tell you N=652?
(2) What do they mean by
MoE = 3.8? (Can you show
how they computed it?)
Fundamental polling result:
Standard error = SE = sqr[p(1-p)/N]
MOE =  1.96  SE
The 95% confidence interval for the
proportion of voters who will vote
for Clinton is
50%  3.8% = [46.2% to 53.8%]
This does not overlap the interval
for Trump, so they would predict
Clinton to win the election (in NH).
The result is not “within the margin
of error.”
Aug.6, 2015. http://www.realclearpolitics.com/epolls/2016/president/nh/new_hampshire_trump_vs_clinton-5596.html
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Part 12: Statistical Inference
Summary
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Methodology: Statistical Inference
Application to credit scoring
Sample statistics as estimators
Point estimation
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Sampling distributions
Confidence intervals
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Sampling variability
The law of large numbers
Unbiasedness and consistency
Proportion
Mean
Using the normal and t distributions instead of the empirical
rule for the width of the interval.
Part 12: Statistical Inference