© 2003 Prentice-Hall, Inc. Chap 6-1
Business Statistics: A First Course
(3rd Edition)
Chapter 6Sampling Distributions
andConfidence Interval
Estimation
© 2003 Prentice-Hall, Inc. Chap 6-2
Chapter Topics
Sampling Distribution of the Mean
Estimation Process Point Estimates Interval Estimates Confidence Interval Estimation for the
Mean ( Known)
Determining Sample Size
© 2003 Prentice-Hall, Inc. Chap 6-3
Chapter Topics
Confidence Interval Estimation for the Mean ( Unknown)
Confidence Interval Estimation for the Proportion
Confidence Interval Estimation and Ethical Issues
(continued)
© 2003 Prentice-Hall, Inc. Chap 6-4
Why Study Sampling Distributions
Sample Statistics are Used to Estimate Population Parameters e.g. estimates the population mean
Problem: Different Samples Provide Different Estimates Large sample gives better estimate; large
sample costs more How good is the estimate?
Approach to Solution: Theoretical Basis is Sampling Distribution
50X
© 2003 Prentice-Hall, Inc. Chap 6-5
Sampling Distribution
Theoretical Probability Distribution of a Sample Statistic
Sample Statistic is a Random Variable Sample mean, sample proportion
Results from Taking All Possible Samples of the Same Size
© 2003 Prentice-Hall, Inc. Chap 6-6
Developing Sampling Distributions
Suppose There is a Population … Population Size N=4 Random Variable, X,
is Age of Individuals Values of X : 18, 20,
22, 24 Measured inYears A
B C
D
© 2003 Prentice-Hall, Inc. Chap 6-7
1
2
1
18 20 22 2421
4
2.236
N
ii
N
ii
X
N
X
N
.3
.2
.1
0 A B C D (18) (20) (22) (24)
Uniform Distribution
P(X)
X
Developing Sampling Distributions
(continued)
Summary Measures for the Population Distribution
© 2003 Prentice-Hall, Inc. Chap 6-8
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
All Possible Samples of Size n=2
16 Samples Taken with Replacement
16 Sample Means1st 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
Developing Sampling Distributions
(continued)
© 2003 Prentice-Hall, Inc. Chap 6-9
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
Sampling Distribution of All Sample Means
18 19 20 21 22 23 240
.1
.2
.3
Sample Means
Distribution
16 Sample Means
Developing Sampling Distributions
(continued)
P X
X
© 2003 Prentice-Hall, Inc. Chap 6-10
1
2
1
2 2 2
18 19 19 2421
16
18 21 19 21 24 211.58
16
N
ii
X
N
i Xi
X
X
N
X
N
Summary Measures of Sampling Distribution
Developing Sampling Distributions
(continued)
© 2003 Prentice-Hall, Inc. Chap 6-11
Comparing the Population with its Sampling
Distribution
18 19 20 21 22 23 240
.1
.2
.3
Sample Means Distribution
n = 2
A B C D (18) (20) (22) (24)
0
.1
.2
.3
PopulationN = 4
21 2.236 21 1.58X X P X P X
X X
© 2003 Prentice-Hall, Inc. Chap 6-12
Properties of Summary Measures
i.e. is unbiased
Standard Error (Standard Deviation) of the Sampling Distribution is Less than the Standard Error of Other Unbiased Estimators
For Sampling with Replacement or without Replacement from Large or Infinite Populations:
As n increases, decreases
X
X
Xn
X
X
© 2003 Prentice-Hall, Inc. Chap 6-13
Unbiasedness
BiasedUnbiased
X X
f X
© 2003 Prentice-Hall, Inc. Chap 6-14
Less Variability
Sampling Distribution of Median Sampling
Distribution of Mean
X
f X
© 2003 Prentice-Hall, Inc. Chap 6-15
Effect of Large Sample
Larger sample size
Smaller sample size
X
f X
© 2003 Prentice-Hall, Inc. Chap 6-16
When the Population is Normal
Central Tendency
Variation
Population Distribution
Sampling Distributions
X
Xn
X50X
4
5X
n
16
2.5X
n
50
10
© 2003 Prentice-Hall, Inc. Chap 6-17
When the Population is Not Normal
Central Tendency
Variation
Population Distribution
Sampling Distributions
X
Xn
X50X
4
5X
n
30
1.8X
n
50
10
© 2003 Prentice-Hall, Inc. Chap 6-18
Central Limit Theorem
As Sample Size Gets Large Enough
Sampling Distribution Becomes Almost Normal Regardless of Shape of Population X
© 2003 Prentice-Hall, Inc. Chap 6-19
How Large is Large Enough?
For Most Distributions, n>30 For Fairly Symmetric Distributions, n>15 For Normal Distribution, the Sampling
Distribution of the Mean is Always Normally Distributed
© 2003 Prentice-Hall, Inc. Chap 6-20
Example:
8 =2 25
7.8 8.2 ?
n
P X
Sampling Distribution
Standardized Normal
Distribution2
.425
X 1Z
8X 8.2 Z0Z
0.5
7.8 8 8.2 87.8 8.2
2 / 25 2 / 25
.5 .5 .3830
X
X
XP X P
P Z
7.8 0.5
.1915
X
© 2003 Prentice-Hall, Inc. Chap 6-21
Estimation Process
Mean, , is unknown
Population Random Sample I am 95%
confident that is between 40 &
60.
Mean X = 50
Sample
© 2003 Prentice-Hall, Inc. Chap 6-22
Point Estimates
Estimate Population
Parameters …
with SampleStatistics
Mean
Proportion
Variance
Difference
p
2
1 2
X
SP
2S
1 2X X
© 2003 Prentice-Hall, Inc. Chap 6-23
Interval Estimates
Provides Range of Values Take into consideration variation in sample
statistics from sample to sample Based on observation from 1 sample Give Information about Closeness to
Unknown Population Parameters Stated in terms of level of confidence
Never 100% sure
© 2003 Prentice-Hall, Inc. Chap 6-24
Confidence Interval Estimates
Mean
Unknown
ConfidenceIntervals
Proportion
Known
© 2003 Prentice-Hall, Inc. Chap 6-25
Confidence Interval for( Known)
Assumptions Population standard deviation is known Population is normally distributed If population is not normal, use large sample
Confidence Interval Estimate
/ 2 / 2X Z X Zn n
© 2003 Prentice-Hall, Inc. Chap 6-26
Elements of Confidence Interval Estimation
Level of Confidence Confidence that the interval will contain the
unknown population parameter Precision (Range)
Closeness to the unknown parameter Cost
Cost required to obtain a sample of size n
© 2003 Prentice-Hall, Inc. Chap 6-27
Level of Confidence
Denoted by A Relative Frequency Interpretation
In the long run, of all the confidence intervals that can be constructed will contain (bracket) the unknown parameter
A Specific Interval Will Either Contain or Not Contain the Parameter No probability involved in a specific interval
100 1 %
100 1 %
© 2003 Prentice-Hall, Inc. Chap 6-28
Interval and Level of Confidence
Confidence Intervals
Intervals extend from
to
of intervals constructed contain ;
do not.
_Sampling Distribution of the Mean
XX Z
X/ 2
/ 2
XX
1
XX Z
1 100%
100 %
/ 2 XZ / 2 XZ
© 2003 Prentice-Hall, Inc. Chap 6-29
Factors Affecting Interval Width
(Precision)
Data Variation Measured by
Sample Size
Level of Confidence
Intervals Extend from
© 1984-1994 T/Maker Co.
Xn
100 1 %
to X XX Z X Z
© 2003 Prentice-Hall, Inc. Chap 6-30
Determining Sample Size (Cost)
Too Big:
• Requires more resources
Too small:
• Won’t do the job
© 2003 Prentice-Hall, Inc. Chap 6-31
Determining Sample Size for Mean
What sample size is needed to be 90% confident of being correct within ± 5? A pilot study suggested that the standard deviation is 45.
Round Up
2 22 2
2 2
1.645 45219.2 220
Error 5
Zn
© 2003 Prentice-Hall, Inc. Chap 6-32
Determining Sample Size for Mean in PHStat
PHStat | Sample Size | Determination for the Mean …
Example in Excel Spreadsheet
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 6-33
Assumptions Population standard deviation is unknown Population is normally distributed If population is not normal, use large sample
Use Student’s t Distribution Confidence Interval Estimate
Confidence Interval for( Unknown)
/ 2, 1 / 2, 1n n
S SX t X t
n n
© 2003 Prentice-Hall, Inc. Chap 6-34
Student’s t Distribution
Zt
0
t (df = 5)
t (df = 13)Bell-ShapedSymmetric
‘Fatter’ Tails
Standard Normal
© 2003 Prentice-Hall, Inc. Chap 6-35
Degrees of Freedom (df )
Number of Observations that are Free to Vary after Sample Mean has been Calculated
Example Mean of 3 numbers is 2
degrees of freedom = n -1 = 3 -1= 2
1
2
3
1 (or any number)
2 (or any number)
3 (cannot vary)
X
X
X
© 2003 Prentice-Hall, Inc. Chap 6-36
Student’s t Table
Upper Tail Area
df .25 .10 .05
1 1.000 3.078 6.314
2 0.817 1.886 2.920
3 0.765 1.638 2.353
t0 2.920t Values
Let: n = 3 df = n - 1 = 2 = .10 /2 =.05
/ 2 = .05
© 2003 Prentice-Hall, Inc. Chap 6-37
Example
/ 2, 1 / 2, 1
8 850 2.0639 50 2.0639
25 2546.69 53.30
n n
S SX t X t
n n
A random sample of 25 has 50 and 8.
Set up a 95% confidence interval estimate for
n X S
© 2003 Prentice-Hall, Inc. Chap 6-38
PHStat | Confidence Interval | Estimate for the Mean, sigma unknown
Example in Excel Spreadsheet
Confidence Interval for( Unknown) in PHStat
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 6-39
Confidence Interval Estimate for Proportion
Assumptions Two categorical outcomes Population follows Binomial distribution Normal approximation can be used if
and Confidence Interval Estimate
5np 1 5n p
/ 2 / 2
1 1S S S SS S
p p p pp Z p p Z
n n
© 2003 Prentice-Hall, Inc. Chap 6-40
ExampleA random sample of 400 Voters showed 32 preferred Candidate A. Set up a 95% confidence interval estimate for p.
/ /
1 1
.08 1 .08 .08 1 .08.08 1.96 .08 1.96
400 400.053 .107
s s s ss s
p p p pp Z p p Z
n n
p
p
© 2003 Prentice-Hall, Inc. Chap 6-41
Confidence Interval Estimate for Proportion in PHStat
PHStat | Confidence Interval | Estimate for the Proportion …
Example in Excel Spreadsheet
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 6-42
Determining Sample Size for Proportion
Out of a population of 1,000, we randomly selected 100 of which 30 were defective. What sample size is needed to be within ± 5% with 90% confidence?
Round Up
2 2
2 2
1 1.645 0.3 0.7
Error 0.05227.3 228
Z p pn
© 2003 Prentice-Hall, Inc. Chap 6-43
Determining Sample Size for Proportion in PHStat
PHStat | Sample Size | Determination for the Proportion …
Example in Excel Spreadsheet
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 6-44
Ethical Issues
Confidence Interval (Reflects Sampling Error) Should Always be Reported Along with the Point Estimate
The Level of Confidence Should Always be Reported
The Sample Size Should be Reported An Interpretation of the Confidence
Interval Estimate Should Also be Provided
© 2003 Prentice-Hall, Inc. Chap 6-45
Chapter Summary
Discussed Sampling Distribution of the Sample Mean
Illustrated Estimation Process Discussed Point Estimates Addressed Interval Estimates Discussed Confidence Interval
Estimation for the Mean ( Known) Addressed Determining Sample Size
© 2003 Prentice-Hall, Inc. Chap 6-46
Chapter Summary
Discussed Confidence Interval Estimation for the Mean ( Unknown)
Discussed Confidence Interval Estimation for the Proportion
Addressed Confidence Interval Estimation and Ethical Issues
(continued)