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Chapter 7Sampling and Point Estimation
Sample This
Chapter 7A
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This Week in Prob & Stat
today
fine print warning: while today’s presentation is mostly conceptual,Thursday’s presentation will be much more mathematical.
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7-1 Introduction
• The field of statistical inference consists of those methods used to make decisions or to draw conclusions about a population.
• These methods utilize the information contained in a sample from the population in drawing conclusions.
• Statistical inference may be divided into two major areas:
• Parameter estimation
• Hypothesis testing
Fundamental Problem: Given that X1, X2, …, Xn is a random samplefrom some unknown population, what can be said about the population?For example, what is the distribution, mean, variance, median, range, etc.
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The Big Picture Again
population
descriptivestatistics
parameter(e.g. mean)
sample
descriptivestatistics
statistic(e.g. sample
mean)
probabilitytheory
deduction
induction (inferential statistics)
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Statistics and Sampling Statistical Inference:
Draw conclusions about a population based on sample. Hypothesis tests and parameter estimation.
Population: Generally impossible or impractical to observe
an entire population. Be aware that population may change over
time. Sample:
A subset of observations from a population. Must be representative of the population. Must be chosen randomly to avoid bias.
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Parameter Estimation
a population
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Estimators
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Sampling – A Pictorial Presentation
X
f(x)Population
Random SampleX1, X2, …, Xn
1
n
ii
XX
n
Sample ( ) XStd dev X
n
2
Xi ~ Population(,2)
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Sampling Distributions
The probability distribution of a statistic is called a sampling distribution.
- Definition makes sense. Statistic is a property of a sample from a population.
- Depends on the population distribution, sample size, and method of sample selection.
- Key statistics are things like the sample mean, variance, proportion, and difference of two means.
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Definition of a Statistic
21
21
1
222
1
ˆˆ
/ˆ
)(1
1ˆ
1ˆ
pp
xx
nxp
xxn
s
xn
x
n
ii
n
ii
Statistic – any function of the observations in a random sample. Examples of point estimates:
A Sampling Distribution is the probability distribution of a statistic.
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Sampling Distributions cont’d
nn
n
n
XXXX
X
X
n
2
2
2222
21
... and
... that so
...
If the Xi have a normal distribution, then so does the sample mean. The Xi are I.I.D.R.V.
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7.2 Sampling Distributions and the Central Limit Theorem
Statistical inference is concerned with making decisions about a population based on the information contained in a random sample from that population.
Definitions:
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The Central Limit Theorem
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Figure 7-1
Distributions of average scores from throwing dice. [Adapted with permission from Box, Hunter, and Hunter (1978).]
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What does it taketo become normal?
Who are you calling
normal?
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More Normalcy
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The CLT in Action
Ten people from a population having a mean weight of 190 lb. with a variance of 400 lb2 get on an elevator having a weight capacity of 2000 lb. What is the probability that their average weight exceeds 200 lb. and they all fall to their death?
2approx 190, 400 /10
200 190Pr 200 Pr Pr 1.581 .0569
/ 20 / 10
x xX n
XX z
n
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Some Normal Thoughts on the Central Limit Theorem (CLT)
CLT tells us that a distribution of means will always be nearly normal if the sample is large enough.
Heights of adults are a result of both genetic and environmental factors. they are polygenic – influenced by many different
genes also many environmental factors; e.g. nutrition and
childhood diseases eventual height is then an average sample from the
large population of “height factors” Height therefore has a normal distribution
The tall and the short of it.
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However All is not Normal
Weights of individuals are not normally distribution It is not just the cumulative result of many small
factors In addition, there may be one or two dominant causes
of obesity; e.g. a glandular disturbance In a similar manner, income is not normally
distributed as a result of a dominant factor – e.g. inherited wealth
On the other hand, mental test scores tend to be normally distributed due to many determinants: e.g. genetics and long-
term environmental conditions
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CLT Revisited
The Main Result (of all time!):
1 2
2
1
If , ,..., is a random sample of size taken
from a population (finite or infinite) with mean and
finite variance , and if Y= , the
limiting form of the distribution of
n
n
ii
X X X n
X
as , is the standard normal distribution.
y
y
YZ
n
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The sum of all Random Variables
A doctor spends an average (mean) of 20 minutes with each patient with a standard deviation of 8 minutes. Today’s appointment book shows 10 patients scheduled this morning (8 – 12).
The good doctor has a luncheon appointment at noon before her afternoon golf outing.
What is the probability she will make the luncheon on time?
10
1
( 200, 640)
240 200Pr{ 240} Pr Pr{ 1.581} .9431
25.3
i y yi
y
y
Y X n
YY Z
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Difference in Sample Means
Approximate Sampling Distribution
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More Mean Differences
1 2
1 2 1 2 1 2
2 22 1 2
1 2 1 2
1 2 1 2
2 21 1 2 2
( )
/ /
y
y
y
y
Y X X
E Y E X X E X E X
V Y V X X V X V Xn n
Y X XZ
n n
2 21 1 1 2 2 2( , ); ( , )X n X n
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A Normal Difference Example
The section 1 class in ENM 661 consisting of 24 students had an average score of 82.7 on their midterm while section 2 consisting of 16 students scored an average of 81.4.
What is the probability that their average scores would differ by at least 1.3 if 1 - 2 = 0. Assume the population standard deviations are known where 1 = 10 and 2 = 12.
1 2 1 2
2 21 1 2 2
82.7 81.4 0( ).358
100 144/ /24 16
Pr .358 .36
X Xz
n n
Z
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Now begins the discussion on point estimation
The discussion on the central limit theorem has now ended.
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Definition of Point Estimate
22 vs., vs. sSorxX
s2 is a population parameter, S2 is a point estimator of s2. The estimate of S2 is s2. S2 has a sampling distribution. But s2 does not – it is just a number.
Point Estimate
is a point estimate of some population parameter of a statistic .
is a point estimator of . After a sample has been selected takes on a particular value .
is a random variable, is not, e.g.
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Properties of Estimators
What makes a good estimator? What is the best estimator for a population parameter?
• Bias - does it hit the target?
• Variance – estimate is based on a sample
• Standard Error and Estimated Standard Error
• Mean Squared Error and Efficiency
• Consistency – how does the estimator behave as the sample size increases?
• Sufficiency – does the estimator use all the information that is available?
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Bias of the Estimator
Def: The point estimator is an unbiased estimator for the parameter if E. If the estimator is not unbiased, then the difference E is called the bias of the estimator .
Is the sample mean unbiased?
1
1 1
1 1[ ] [ ]
n
i n ni
i ii i
Xn
E X E E X E Xn n n n
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7-3 General Concepts of Point Estimation
7-3.1 Unbiased Estimators
Definition
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Example 7-1
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Example 7-1 (continued)
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A Biased Estimator
Define an estimator for the population variance to be:
2 2 2 2
1 1
2 2 2 2 2
1 1
22 2
1
22 2 2
22 2
1 1( )
1
n n
i ii i
n n
i ii i
S X X X nXn n
E nS E X nX E X E nX
nE X nn
nn
nn n
n
2 2 2using: [ ]E X
22 2 2
2 2 2
1
1
nE S
n n
nE S E S
n
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7-3.2 Variance of a Point Estimator
Definition
Figure 7-5 The sampling distributions of two unbiased estimators
.ˆˆ21 and
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Variance of Estimator
Sample mean is the MVUE for the population mean for a population with normal distribution.
Generally, the stat package you use is making the reasonable choices for you.
Example of bad choice: sample size n=2Method 1: estimate mean as (X1 + X2)/2Method 2: estimate mean as (X1 + 2X2)/3
Variance of method 1 is 2/2 Variance of method 2 is 52/9
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7-3.2 Variance of a Point Estimator
I just knew it was going to be the sample mean.
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BLUE Estimator
Best Linear Unbiased Estimator (BLUE) Best is defined as the minimum
variance estimator from among all unbiased linear estimators
Is the sample mean a BLUE estimator for the population mean?
1
n
ii
XX
n
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An Engineering Management Bonus Round!!!!
A real world example of sampling, parameter
estimation, and fishing for the correct answer.
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How many Fish are in the Lake?
Let N = the number of fish in the lake k = the number of fish caught and tagged, and
released back into the lake allow for the tagged fish to be uniformly dispersed
within the lake X = a RV, the number of tagged fish caught in the
follow-on sample of size n Then an estimate for the number of fish in the lake
is found by assuming
ˆThen
k X
N nk n
NX
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Is N-hat unbiased, BLUE, or MVUE?
counting the fishin the lake
2
ˆ ;
1ˆ
1ˆ
k nN
Xk n
E N E kn EX X
k nVar N Var kn Var
X X
, ,X hypergeometric N n k
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Point Estimation- to be continued
next time- making standard errors- those magic moments- maximizing likelihoods
ENM 500 students engaged in random sampling.
ENM 500 students caughtdiscussing the central limittheorem.