statistics for economics · econ 509, by dr. m. zainal ch. 6-14. ch. 6-15 the standard normal table...
TRANSCRIPT
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Statistics for Economics
Dr. Mohammad Zainal
Chapter 6Sampling and
Sampling Distributions
ECON 509
Department of Economics
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Chapter Goals
After completing this chapter, you should be able to:
Describe a simple random sample and why sampling is
important
Explain the difference between descriptive and
inferential statistics
Define the concept of a sampling distribution
Determine the mean and standard deviation for the
sampling distribution of the sample mean,
Describe the Central Limit Theorem and its importance
Determine the mean and standard deviation for the
sampling distribution of the sample proportion,
Describe sampling distributions of sample variances
p̂
X
ECON 509, by Dr. M. Zainal Chap 6-2
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The Normal Distribution
Descriptive statistics
Collecting, presenting, and describing data
Inferential statistics
Drawing conclusions and/or making decisions
concerning a population based only on
sample data
6.0
ECON 509, by Dr. M. Zainal Chap 6-3
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Probability Distributions
Continuous
Probability
Distributions
Binomial
Hypergeometric
Poisson
Probability
Distributions
Discrete
Probability
Distributions
Normal
Uniform
Exponential
.
.
.
.
ECON 509, by Dr. M. Zainal Chap 6-4
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Ch. 6-5
Continuous Probability Distributions
A continuous random variable is a variable that
can assume any value on a continuum (can
assume an uncountable number of values)
thickness of an item
time required to complete a task
temperature of a solution
height, in inches
These can potentially take on any value,
depending only on the ability to measure
accurately.
ECON 509, by Dr. M. Zainal
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Ch. 6-6
The Normal Distribution
‘Bell Shaped’
Symmetrical
Mean, Median and Modeare Equal
Location is determined by the mean, μ
Spread is determined by the standard deviation, σ
The random variable has an infinite theoretical range: + to
Mean
= Median
= Mode
x
f(x)
μ
σ
ECON 509, by Dr. M. Zainal
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Ch. 6-7
The Normal Distribution Shape
x
f(x)
μ
σ
Changing μ shifts the
distribution left or right.
Changing σ increases
or decreases the
spread.
ECON 509, by Dr. M. Zainal
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Ch. 6-8
The Normal Distribution Shape
By varying the parameters μ and σ, we obtain
different normal distributions
ECON 509, by Dr. M. Zainal
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Ch. 6-9
Finding Normal Probabilities
a b x
f(x) P a x b( )
Probability is measured by the area
under the curve
ECON 509, by Dr. M. Zainal
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ECON 509, by Dr. M. Zainal Ch. 6-10
f(x)
xμ
Probability as Area Under the Curve
0.50.5
The total area under the curve is 1.0, and the curve is
symmetric, so half is above the mean, half is below
1.0)xP(
0.5)xP(μ 0.5μ)xP(
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Ch. 6-11
The Standard Normal Distribution
Also known as the “z” distribution
Mean is defined to be 0
Standard Deviation is 1
z
f(z)
0
1
Values above the mean have positive z-values,
values below the mean have negative z-values
ECON 509, by Dr. M. Zainal
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Ch. 6-12
The Standard Normal
Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normaldistribution (z)
Need to transform x units into z units
σ
μxz
ECON 509, by Dr. M. Zainal
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Ch. 6-13
Example
If x is distributed normally with mean of 100
and standard deviation of 50, the z value for
x = 250 is
This says that x = 250 is three standard
deviations (3 increments of 50 units) above
the mean of 100.
3.050
100250
σ
μxz
ECON 509, by Dr. M. Zainal
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Comparing x and z units
z
100
3.00
250 x
Note that the distribution is the same, only the
scale has changed. We can express the problem in
original units (x) or in standardized units (z)
μ = 100
σ = 50
ECON 509, by Dr. M. Zainal Ch. 6-14
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Ch. 6-15
The Standard Normal Table
The Standard Normal table in the textbook
gives the probability from the mean (zero)
up to a desired value for z
z0 2.00
.4772Example:
P(0 < z < 2.00) = .4772
ECON 509, by Dr. M. Zainal
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The Standard Normal Table
The value within the
table gives the
probability from z = 0
up to the desired z
value
z 0.00 0.01 0.02 …
0.1
0.2
.4772
2.0P(0 < z < 2.00) = .4772
The row shows
the value of z
to the first
decimal point
The column gives the value of
z to the second decimal point
2.0
.
.
.
(continued)
ECON 509, by Dr. M. Zainal Chap 6-16
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General Procedure for Finding Probabilities
Draw the normal curve for the problem in
terms of x
Translate x-values to z-values
Use the Standard Normal Table
To find P(a < x < b) when x is distributed
normally:
ECON 509, by Dr. M. Zainal Chap 6-17
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Z Table example
Suppose x is normal with mean 8.0 and
standard deviation 5.0. Find P(8 < x < 8.6)
P(8 < x < 8.6)
= P(0 < z < 0.12)
Z0.120
x8.68
05
88
σ
μxz
0.125
88.6
σ
μxz
Calculate z-values:
ECON 509, by Dr. M. Zainal Chap 6-18
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Z Table example
Suppose x is normal with mean 8.0 and
standard deviation 5.0. Find P(8 < x < 8.6)
P(0 < z < 0.12)
z0.120x8.68
P(8 < x < 8.6)
= 8
= 5
= 0
= 1
(continued)
ECON 509, by Dr. M. Zainal Chap 6-19
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ECON 509, by Dr. M. Zainal Chap 6-20
Z
0.12
z .00 .01
0.0 .0000 .0040 .0080
.0398 .0438
0.2 .0793 .0832 .0871
0.3 .1179 .1217 .1255
Solution: Finding P(0 < z < 0.12)
.0478.02
0.1 .0478
Standard Normal Probability
Table (Portion)
0.00
= P(0 < z < 0.12)
P(8 < x < 8.6)
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ECON 509, by Dr. M. Zainal
Finding Normal Probabilities
Suppose x is normal with mean 8.0
and standard deviation 5.0.
Now Find P(x < 8.6)
Z
8.6
8.0
Chap 6-21
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ECON 509, by Dr. M. Zainal
Finding Normal Probabilities
Suppose x is normal with mean 8.0
and standard deviation 5.0.
Now Find P(x < 8.6)
(continued)
Z
0.12
.0478
0.00
.5000P(x < 8.6)
= P(z < 0.12)
= P(z < 0) + P(0 < z < 0.12)
= .5 + .0478 = .5478
Chap 6-22
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ECON 509, by Dr. M. Zainal
Upper Tail Probabilities
Suppose x is normal with mean 8.0
and standard deviation 5.0.
Now Find P(x > 8.6)
Z
8.6
8.0
Chap 6-23
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ECON 509, by Dr. M. Zainal
Now Find P(x > 8.6)…
(continued)
Z
0.12
0Z
0.12
.0478
0
.5000 .50 - .0478
= .4522
P(x > 8.6) = P(z > 0.12) = P(z > 0) - P(0 < z < 0.12)
= .5 - .0478 = .4522
Upper Tail Probabilities
Chap 6-24
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ECON 509, by Dr. M. Zainal Chap 6-25
Lower Tail Probabilities
Suppose x is normal with mean 8.0
and standard deviation 5.0.
Now Find P(7.4 < x < 8)
Z
7.48.0
-
ECON 509, by Dr. M. Zainal
Lower Tail Probabilities
Now Find P(7.4 < x < 8)…
Z
7.48.0
The Normal distribution is
symmetric, so we use the
same table even if z-values
are negative:
P(7.4 < x < 8)
= P(-0.12 < z < 0)
= .0478
(continued)
.0478
Chap 6-26
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ECON 509, by Dr. M. Zainal
Z Table example
Example: Find the following areas under the
standard normal curve.
a) P(0 < z < 5.65)
b) P( z < - 5.3)
Chap 6-27
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Z Table example
Example: The lifetime of a calculator
manufactured by a company has a normal
distribution with a mean of 54 months and a
standard deviation of 8 months. The company
guarantees that any calculator that starts
malfunctioning within 36 months of the purchase
will be replaced by a new one. What percentage
of such calculators are expected to be
replaced?ECON 509, by Dr. M. Zainal Chap 6-28
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Determining the z and x values
We reverse the procedure of finding the area
under the normal curve for a specific value of z
or x to finding a specific value of z or x for a
known area under the normal curve.
z 0.00 0.01 0.02 …
0.1
0.2
.47722.0
.
.
.
ECON 509, by Dr. M. Zainal Chap 6-29
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Determining the z and x values
Example: Find a point z such that the area
under the standard normal curve between 0 and
z is .4251 and the value of z is positive
ECON 509, by Dr. M. Zainal Chap 6-30
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Finding x for a normal dist.
To find an x value when an area under a normal
distribution curve is given, we do the following
▪ Find the z value corresponding to that x value
from the standard normal curve.
▪ Transform the z value to x by substituting the
values of , , and z in the following formula
x z
ECON 509, by Dr. M. Zainal Chap 6-31
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Finding x for a normal dist.
Example: Recall the calculators example, it is
known that the life of a calculator manufactured
by a factory has a normal distribution with a
mean of 54 months and a standard deviation of
8 months. What should the warranty period be
to replace a malfunctioning calculator if the
company does not want to replace more than
0.5 % of all the calculators sold?
ECON 509, by Dr. M. Zainal Chap 6-32
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Tools of Business Statistics
Descriptive statistics
Collecting, presenting, and describing data
Inferential statistics
Drawing conclusions and/or making decisions
concerning a population based only on
sample data
6.1
ECON 509, by Dr. M. Zainal Chap 6-33
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Populations and Samples
A Population is the set of all items or individuals of interest
Examples: All likely voters in the next election
All parts produced today
All sales receipts for November
A Sample is a subset of the population
Examples: 1000 voters selected at random for interview
A few parts selected for destructive testing
Random receipts selected for audit
Ch. 6-34ECON 509, by Dr. M. Zainal
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Population vs. Sample
Ch. 6-35
a b c d
ef gh i jk l m n
o p q rs t u v w
x y z
Population Sample
b c
g i n
o r u
y
ECON 509, by Dr. M. Zainal
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Why Sample?
Less time consuming than a census
Less costly to administer than a census
It is possible to obtain statistical results of a
sufficiently high precision based on samples.
Ch. 6-36ECON 509, by Dr. M. Zainal
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Simple Random Samples
Every object in the population has an equal chance of
being selected
Objects are selected independently
Samples can be obtained from a table of random
numbers or computer random number generators
A simple random sample is the ideal against which
other sample methods are compared
Ch. 6-37ECON 509, by Dr. M. Zainal
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Inferential Statistics
Making statements about a population by
examining sample results
Sample statistics Population parameters
(known) Inference (unknown, but can
be estimated from
sample evidence)
Ch. 6-38
SamplePopulation
ECON 509, by Dr. M. Zainal
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Inferential Statistics
Estimation
e.g., Estimate the population mean
weight using the sample mean
weight
Hypothesis Testing
e.g., Use sample evidence to test
the claim that the population mean
weight is 120 pounds
Ch. 6-39
Drawing conclusions and/or making decisions concerning a population based on sample results.
ECON 509, by Dr. M. Zainal
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Sampling Distributions
A sampling distribution is a distribution of
all of the possible values of a statistic for
a given size sample selected from a
population
Ch. 6-40
6.2
ECON 509, by Dr. M. Zainal
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Chapter Outline
Ch. 6-41
Sampling
Distributions
Sampling
Distribution of
Sample
Mean
Sampling
Distribution of
Sample
Proportion
Sampling
Distribution of
Sample
Variance
ECON 509, by Dr. M. Zainal
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Sampling Distributions ofSample Means
Ch. 6-42
Sampling
Distributions
Sampling
Distribution of
Sample
Mean
Sampling
Distribution of
Sample
Proportion
Sampling
Distribution of
Sample
Variance
ECON 509, by Dr. M. Zainal
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Developing a Sampling Distribution
Assume there is a population …
Population size N=4
Random variable, X,
is age of individuals
Values of X:
18, 20, 22, 24 (years)
Ch. 6-43
A B CD
ECON 509, by Dr. M. Zainal
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Developing a Sampling Distribution
Ch. 6-44
.25
018 20 22 24
A B C D
Uniform Distribution
P(x)
x
(continued)
Summary Measures for the Population Distribution:
214
24222018
N
Xμ i
2.236N
μ)(Xσ
2
i
ECON 509, by Dr. M. Zainal
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Now consider all possible samples of size n = 2
Ch. 6-45
1st 2nd Observation Obs 18 20 22 24
18 18,18 18,20 18,22 18,24
20 20,18 20,20 20,22 20,24
22 22,18 22,20 22,22 22,24
24 24,18 24,20 24,22 24,24
16 possible samples
(sampling with
replacement)
1st 2nd Observation
Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
(continued)
Developing a Sampling Distribution
16 Sample
Means
ECON 509, by Dr. M. Zainal
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Sampling Distribution of All Sample Means
Ch. 6-46
1st 2nd Observation
Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
18 19 20 21 22 23 240
.1
.2
.3
P(X)
X
Sample Means
Distribution16 Sample Means
_
Developing a Sampling Distribution
(continued)
(no longer uniform)
_
ECON 509, by Dr. M. Zainal
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Summary Measures of this Sampling Distribution:
Ch. 6-47
Developing aSampling Distribution
(continued)
μ2116
24211918
N
X)XE( i
1.5816
21)-(2421)-(1921)-(18
N
μ)X(σ
222
2i
X
ECON 509, by Dr. M. Zainal
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Comparing the Population with its Sampling Distribution
Ch. 6-48
18 19 20 21 22 23 240
.1
.2
.3 P(X)
X18 20 22 24
A B C D
0
.1
.2
.3
Population
N = 4
P(X)
X_
1.58σ 21μXX2.236σ 21μ
Sample Means Distributionn = 2
_
ECON 509, by Dr. M. Zainal
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Expected Value of Sample Mean
Let X1, X2, . . . Xn represent a random sample from a
population
The sample mean value of these observations is
defined as
Ch. 6-49
n
1i
iXn
1X
ECON 509, by Dr. M. Zainal
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Standard Error of the Mean
Different samples of the same size from the same
population will yield different sample means
A measure of the variability in the mean from sample to
sample is given by the Standard Error of the Mean:
Note that the standard error of the mean decreases as
the sample size increases
Ch. 6-50
n
σσ
X
ECON 509, by Dr. M. Zainal
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If sample values are not independent
If the sample size n is not a small fraction of the
population size N, then individual sample members
are not distributed independently of one another
Thus, observations are not selected independently
A correction is made to account for this:
or
Ch. 6-51
(continued)
1N
nN
n
σσ
X
1N
nN
n
σ)XVar(
2
ECON 509, by Dr. M. Zainal
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If the Population is Normal
If a population is normal with mean μ and
standard deviation σ, the sampling distribution
of is also normally distributed with
and
If the sample size n is not large relative to the population size N, then
and
Ch. 6-52
X
μμX
n
σσ
X
1N
nN
n
σσ
X
μμ
X
ECON 509, by Dr. M. Zainal
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Z-value for Sampling Distributionof the Mean
Z-value for the sampling distribution of :
Ch. 6-53
where: = sample mean
= population mean
= standard error of the mean
Xμ
Xσ
μ)X(Z
X
xσ
ECON 509, by Dr. M. Zainal
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Sampling Distribution Properties
(i.e. is unbiased )
Ch. 6-54
Normal Population
Distribution
Normal Sampling
Distribution
(has the same mean)
xx
x
μμx
μ
xμ
ECON 509, by Dr. M. Zainal
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Sampling Distribution Properties
For sampling with replacement:
As n increases,
decreases
Ch. 6-55
Larger
sample size
Smaller
sample size
x
(continued)
xσ
μECON 509, by Dr. M. Zainal
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If the Population is not Normal
We can apply the Central Limit Theorem:
Even if the population is not normal,
…sample means from the population will beapproximately normal as long as the sample size is large enough.
Properties of the sampling distribution:
and
Ch. 6-56
μμ x n
σσ x
ECON 509, by Dr. M. Zainal
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Central Limit Theorem
Ch. 6-57
n↑As the
sample
size gets
large
enough…
the sampling
distribution
becomes
almost normal
regardless of
shape of
population
xECON 509, by Dr. M. Zainal
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If the Population is not Normal
Ch. 6-58
Population Distribution
Sampling Distribution
(becomes normal as n increases)
Central Tendency
Variation
x
x
Larger
sample
size
Smaller
sample size
(continued)
Sampling distribution
properties:
μμ x
n
σσ x
xμ
μ
ECON 509, by Dr. M. Zainal
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How Large is Large Enough?
For most distributions, n > 25 will give a
sampling distribution that is nearly normal
For normal population distributions, the
sampling distribution of the mean is always
normally distributed
Ch. 6-59ECON 509, by Dr. M. Zainal
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Example
Suppose a large population has mean μ = 8
and standard deviation σ = 3. Suppose a
random sample of size n = 36 is selected.
What is the probability that the sample mean is
between 7.8 and 8.2?
Ch. 6-60ECON 509, by Dr. M. Zainal
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Example
Solution:
Even if the population is not normally
distributed, the central limit theorem can be
used (n > 25)
… so the sampling distribution of is
approximately normal
… with mean = 8
…and standard deviation
Ch. 6-61
(continued)
x
xμ
0.536
3
n
σσ x
ECON 509, by Dr. M. Zainal
-
Example
Solution (continued):
Ch. 6-62
(continued)
0.38300.5)ZP(-0.5
363
8-8.2
nσ
μ- μ
363
8-7.8P 8.2) μ P(7.8
X
X
Z7.8 8.2-0.5 0.5
Sampling
Distribution
Standard Normal
Distribution .1915
+.1915
Population
Distribution
??
??
????
????
Sample Standardize
8μ 8μX 0μz xX
ECON 509, by Dr. M. Zainal
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Acceptance Intervals
Goal: determine a range within which sample means are
likely to occur, given a population mean and variance
By the Central Limit Theorem, we know that the distribution of X
is approximately normal if n is large enough, with mean μ and
standard deviation
Let zα/2 be the z-value that leaves area α/2 in the upper tail of the
normal distribution (i.e., the interval - zα/2 to zα/2 encloses
probability 1 – α)
Then
is the interval that includes X with probability 1 – α
Ch. 6-63
Xσ
X/2σzμ
ECON 509, by Dr. M. Zainal
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Sampling
Distributions
Sampling
Distribution of
Sample
Mean
Sampling
Distribution of
Sample
Proportion
Sampling
Distribution of
Sample
Variance
Sampling Distributions of Sample Proportions
Ch. 6-64
6.3
ECON 509, by Dr. M. Zainal
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Sampling Distributions of Sample Proportions
P = the proportion of the population having some characteristic
Sample proportion ( ) provides an estimateof P:
0 ≤ ≤ 1
has a binomial distribution, but can be approximated by a normal distribution when nP(1 – P) > 5
Ch. 6-65
size sample
interest ofstic characteri the having sample the in itemsofnumber
n
Xp ˆ
p̂
p̂
p̂
ECON 509, by Dr. M. Zainal
-
Sampling Distribution of p
Normal approximation:
Properties:
and
Ch. 6-66
(where P = population proportion)
Sampling Distribution
.3
.2
.1
00 . 2 .4 .6 8 1
P)pE( ˆn
P)P(1
n
XVarσ2
p
ˆ
^
)PP( ˆ
P̂
ECON 509, by Dr. M. Zainal
-
Z-Value for Proportions
Ch. 6-67
n
P)P(1
Pp
σ
PpZ
p
ˆˆ
ˆ
Standardize to a Z value with the formula:p̂
ECON 509, by Dr. M. Zainal
-
Example
If the true proportion of voters who support
Proposition A is P = .4, what is the probability
that a sample of size 200 yields a sample
proportion between .40 and .45?
Ch. 6-68
▪ i.e.: if P = .4 and n = 200, what is
P(.40 ≤ ≤ .45) ?p̂
ECON 509, by Dr. M. Zainal
-
Example
if P = .4 and n = 200, what is
P(.40 ≤ ≤ .45) ?
Ch. 6-69
(continued)
.03464200
.4).4(1
n
P)P(1σ
p
ˆ
1.44)ZP(0
.03464
.40.45Z
.03464
.40.40P.45)pP(.40
ˆ
Find :
Convert to
standard
normal:
pσ ˆ
p̂
ECON 509, by Dr. M. Zainal
-
Example
if P = .4 and n = 200, what is
P(.40 ≤ ≤ .45) ?
Ch. 6-70
Z.45 1.44
.4251
Standardize
Sampling DistributionStandardized
Normal Distribution
(continued)
Use standard normal table: P(0 ≤ Z ≤ 1.44) = .4251
.40 0p̂
p̂
ECON 509, by Dr. M. Zainal
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Sampling Distributions ofSample Variance
Ch. 6-71
Sampling
Distributions
Sampling
Distribution of
Sample
Mean
Sampling
Distribution of
Sample
Proportion
Sampling
Distribution of
Sample
Variance
6.4
ECON 509, by Dr. M. Zainal
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Sample Variance
Let x1, x2, . . . , xn be a random sample from a
population. The sample variance is
the square root of the sample variance is called
the sample standard deviation
the sample variance is different for different
random samples from the same population
Ch. 6-72
n
1i
2
i
2 )x(x1n
1s
ECON 509, by Dr. M. Zainal
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Sampling Distribution ofSample Variances
The sampling distribution of s2 has mean σ2
If the population distribution is normal, then
If the population distribution is normal then
has a 2 distribution with n – 1 degrees of freedom
Ch. 6-73
22 σ)E(s
1n
2σ)Var(s
42
2
2
σ
1)s-(n
ECON 509, by Dr. M. Zainal
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The Chi-square Distribution
The chi-square distribution is a family of
distributions, depending on degrees of freedom
(Like the t distribution).
The chi-square distribution curve starts at the
origin and lies entirely to the right of the vertical
axis.
The chi-square distribution assumes
nonnegative values only, and these are denoted
by the symbol 2 (read as “chi-square”).
ECON 509, by Dr. M. Zainal Ch. 6-74
-
The Chi-square Distribution
The shape of a specific chi-square distribution
depends on the number of degrees of freedom.
peak of a 2 distribution curve with 1 or 2
degrees of freedom occurs at zero and for a
curve with 3 or more degrees of freedom at
(df−2).
0 4 8 12 16 20 24 28 0 4 8 12 16 20 24 28 0 4 8 12 16 20 24 28
d.f. = 1 d.f. = 5 d.f. = 15
2 22
ECON 509, by Dr. M. Zainal Ch. 6-75
-
Degrees of Freedom (df)
Idea: Number of observations that are free to varyafter sample mean has been calculated
Example: Suppose the mean of 3 numbers is 8.0
Let X1 = 7
Let X2 = 8
What is X3?
Ch. 6-76
If the mean of these three
values is 8.0,
then X3 must be 9
(i.e., X3 is not free to vary)
Here, n = 3, so degrees of freedom = n – 1 = 3 – 1 = 2
(2 values can be any numbers, but the third is not free to vary
for a given mean)ECON 509, by Dr. M. Zainal
-
Finding the Critical Value
The critical value, , is found from the
chi-square table
2
2
2
ECON 509, by Dr. M. Zainal Ch. 6-77
-
Finding the Critical Value
ECON 509, by Dr. M. Zainal Ch. 6-78
-
Finding the Critical Value
Example: Find the value of 2 for 7 degrees of freedom and an
area of .10 in the right tail of the chi-square distribution curve.
ECON 509, by Dr. M. Zainal Ch. 6-79
-
Finding the Critical Value
Example: Find the value of 2 for 9 degrees of freedom and an
area of .05 in the left tail of the chi-square distribution curve.
ECON 509, by Dr. M. Zainal Ch. 6-80
-
Chi-square Example
A commercial freezer must hold a selected
temperature with little variation. Specifications call
for a standard deviation of no more than 4 degrees
(a variance of 16 degrees2).
Ch. 6-81
▪ A sample of 14 freezers is to be
tested
▪ What is the upper limit (K) for the
sample variance such that the
probability of exceeding this limit,
given that the population standard
deviation is 4, is less than 0.05?
ECON 509, by Dr. M. Zainal
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Finding the Chi-square Value
Use the chi-square distribution with area 0.05 in the upper tail:
Ch. 6-82
probability
α = .05
213
2
213
= 22.36
= 22.36 (α = .05 and 14 – 1 = 13 d.f.)
2
22
σ
1)s(nχ
Is chi-square distributed with (n – 1) = 13
degrees of freedom
ECON 509, by Dr. M. Zainal
-
Chi-square Example
Ch. 6-83
0.0516
1)s(nPK)P(s 213
22
χSo:
(continued)
213 = 22.36 (α = .05 and 14 – 1 = 13 d.f.)
22.3616
1)K(n
(where n = 14)
so 27.521)(14
)(22.36)(16K
If s2 from the sample of size n = 14 is greater than 27.52, there is
strong evidence to suggest the population variance exceeds 16.
or
ECON 509, by Dr. M. Zainal
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Chapter Summary
Introduced sampling distributions
Described the sampling distribution of sample means
For normal populations
Using the Central Limit Theorem
Described the sampling distribution of sample
proportions
Introduced the chi-square distribution
Examined sampling distributions for sample variances
Calculated probabilities using sampling distributions
Ch. 6-84ECON 509, by Dr. M. Zainal
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Copyright
The materials of this presentation were mostly
taken from the PowerPoint files accompanied
Business Statistics: A Decision-Making
Approach, 7e © 2008 Prentice-Hall, Inc.
ECON 509, by Dr. M. Zainal Chap 6-85