Chap 7-1
Basic Business Statistics(10th Edition)
Chapter 7
Sampling Distributions
Chap 7-2
Chapter Topics
Sampling Distribution of the Mean
The Central Limit Theorem
Sampling Distribution of the Proportion
Sampling from Finite Population
Chap 7-3
Why Study SamplingDistributions
Sample Statistics are Used to EstimatePopulation Parameters
E.g., estimates the population mean
Problem: Different Samples Provide DifferentEstimates
Large sample gives better estimate; large samplecosts more
How good is the estimate?
Approach to Solution: Theoretical Basis isSampling Distribution
Chap 7-4
Sampling Distribution
Theoretical Probability Distribution of aSample Statistic
Sample Statistic is a Random Variable
Sample mean, sample proportion
Results from Taking All Possible Samples ofthe Same Size
Chap 7-5
Developing SamplingDistributions
Suppose There is a Population …
Population Size N=4
Random Variable, X,is Age of IndividualsMeasured in Years
Values of X : 18, 20,22, 24
A
B
C
D
Chap 7-6
.3
.2
.1
0
A B C D
(18) (20) (22) (24)
Uniform Distribution
P(X)
X
Developing SamplingDistributions
(continued)
Summary Measures for the Population Distribution
© 2004 Prentice-Hall, Inc.
Chap 7-7
All Possible Samples of Size n=2
Nn = 42 = 16Samples Taken withReplacement
16 Sample Means
Developing SamplingDistributions
(continued)
© 2004 Prentice-Hall, Inc.
Chap 7-8
Sampling Distribution of All Sample Means
18 19 20 21 22 23 24
0
.1
.2
.3
X
Sample MeansDistribution
16 Sample Means
_
Developing SamplingDistributions
(continued)
Chap 7-9
Summary Measures of Sampling Distribution
Developing SamplingDistributions
(continued)
Chap 7-10
Comparing the Population withIts Sampling Distribution
18 19 20 21 22 23 24
0
.1
.2
.3
X
Sample Means Distribution
n = 2
A B C D
(18) (20) (22) (24)
0
.1
.2
.3
Population
N = 4
X
_
Chap 7-11
Properties of Summary Measures
I.e., is unbiased
Standard Error (Standard Deviation) of theSampling Distribution is Less Than theStandard Error of Other Unbiased Estimators
For Sampling with Replacement or withoutReplacement from Large or Infinite Populations:
As n increases, decreases
Chap 7-12
Unbiasedness ( )
Unbiased
Chap 7-13
Less Variability
SamplingDistributionof Median
SamplingDistribution ofMean
Standard Error (Standard Deviation) of theSampling Distribution is Less Than theStandard Error of Other Unbiased Estimators
Chap 7-14
Effect of Large Sample
Largersample size
Smallersample size
Chap 7-15
When the Population is Normal
Central Tendency
Variation
Population Distribution
Sampling Distributions
Chap 7-16
When the Population isNot Normal
Central Tendency
Variation
Population Distribution
Sampling Distributions
Chap 7-17
Central Limit Theorem
As SampleSize GetsLargeEnough
SamplingDistributionBecomesAlmostNormalRegardlessof Shape ofPopulation
Chap 7-18
How Large is Large Enough?
For Most Distributions, n>30
For Fairly Symmetric Distributions, n>15
For Normal Distribution, the SamplingDistribution of the Mean is Always NormallyDistributed Regardless of the Sample Size
This is a property of sampling from a normalpopulation distribution and is NOT a result of thecentral limit theorem
Chap 7-19
Example:
Sampling Distribution
StandardizedNormal Distribution
Chap 7-20
Population Proportion
Categorical Variable
E.g., Gender, Voted for Bush, College Degree
Proportion of Population Having aCharacteristic
Sample Proportion Provides an Estimate
If Two Outcomes, X Has a BinomialDistribution
Possess or do not possess characteristic
Chap 7-21
Sampling Distribution ofSample Proportion
Approximated byNormal Distribution
Mean:
Standard error:
p = population proportion
Sampling Distribution
f(ps)
.3
.2
.1
0
0 . 2 .4 .6 8 1
ps
Chap 7-22
Standardizing SamplingDistribution of Proportion
Sampling Distribution
StandardizedNormal Distribution
Chap 7-23
Example:
Sampling Distribution
StandardizedNormal Distribution
Chap 7-24
Sampling from Finite Population(CD ROM Topic)
Modify Standard Error if Sample Size (n) isLarge Relative to Population Size (N )
Use Finite Population Correction Factor (FPC)
Standard Error with FPC
Chap 7-25
Chapter Summary
Discussed Sampling Distribution of the SampleMean
Described the Central Limit Theorem
Discussed Sampling Distribution of the SampleProportion
Described Sampling from Finite Populations