Chapter 8 Confidence Interval Estimation Estimation Process Point Estimates Interval Estimates Confidence Interval Estimation for the Mean ( Known ) Confidence Interval Estimation for the Mean ( Unknown ) Confidence Interval Estimation for the Proportion

1. © 2003 Prentice-Hall, Inc. Chap 8-1
Basic Business Statistics
(9th
Edition)
Chapter 8
Confidence Interval Estimation

2. © 2003 Prentice-Hall, Inc.
Chap 8-2
Chapter Topics
 Estimation Process
 Point Estimates
 Interval Estimates
 Confidence Interval Estimation for the Mean
( Known )
 Confidence Interval Estimation for the Mean
( Unknown )
 Confidence Interval Estimation for the Proportion
σ
σ

3. © 2003 Prentice-Hall, Inc.
Chap 8-3
Examples of where CI are is used
(continued)
CI of Means
1000 Salaries of people in Phnom Penh:
Sample Mean =
How confident are you this is the same as the Population Mean
Salary of the Phnom Pehn? (The average if you had all the data
for every one in Phnom Phen)
Two Methods to work this out: -
1. Confidence Interval Estimation for the Mean ( Known )
You could use the Standard Deviation of men in Cambodia
2. Confidence Interval Estimation for the Mean ( Unknown )
σ
σ

4. © 2003 Prentice-Hall, Inc.
Chap 8-4
Examples of CI for Proportions
 How many of you are satisfied with the with the
the Streets are cleaned in Phnom Penh?
 10 People
 Number who say Yes = Number who say No =
 Proportion = No/(Total who Answered)
 How confident are you this represents the whole
of the class?

5. © 2003 Prentice-Hall, Inc.
Chap 8-5
Estimation Process
Mean, µ, is
unknown
Population Random
Sample I am 95%
confident that µ
is between 40 &
60.
Mean
X = 50
Sample

6. © 2003 Prentice-Hall, Inc.
Chap 8-6
Point Estimates
Estimate Population
Parameters …
with Sample
Statistics
Mean
Proportion
Variance
Difference
µ
p
2
σ
1 2µ µ−
X
SP
2
S
1 2X X−

7. © 2003 Prentice-Hall, Inc.
Chap 8-7
Interval Estimates
 Provide 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

8. © 2003 Prentice-Hall, Inc.
Chap 8-8
Confidence Interval Estimates
Mean
σ Unknown
Confidence
Intervals
Proportion
σ Known

9. © 2003 Prentice-Hall, Inc.
Chap 8-9
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

 is called the sampling error or
margin of error
µ
σ
/ 2 /2X Z X Z
n n
α α
σ σ
µ− ≤ ≤ +
Standard Error
Critical Value
/ 2e Z
n
α
σ
=

10. © 2003 Prentice-Hall, Inc.
Chap 8-10
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

11. © 2003 Prentice-Hall, Inc.
Chap 8-11
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 %α−
( )100 1 %α−

12. © 2003 Prentice-Hall, Inc.
Chap 8-12
Interval and Level of Confidence
Confidence
Intervals
extend from
to
of intervals
constructed
contain ;
do
not.
_Sampling Distribution of the Mean
X
X Zσ−
X
σ
/ 2α
/ 2α
X
X
µ µ=
1 α−
X
X Zσ+
( )1 100%α−
µ
100 %α
/ 2 X
Zαµ σ+/ 2 X
Zαµ σ−

13. © 2003 Prentice-Hall, Inc.
Chap 8-13
Step 3 :Z is found by
looking Table Value on Page
811
How to find the Z Value based on
a Confidence Interval (CI)
/ 2α
/ 2α
X
X
µ µ=
1 α−
/ 2 X
Zαµ σ+
/ 2αTable
Value
Table Value = 1- / 2α
/2α
Z /2α
Step 1. Find / 2α
)
100
CI
(−1=α
Step 2. Find Table Value
Step 4 :Find CI
/ 2 / 2X Z X Z
n n
α α
σ σ
µ− ≤ ≤ +

14. © 2003 Prentice-Hall, Inc.
Chap 8-14
Z Values for Typical Confidence Intervals (CI)
90 1.64
95 1.96
99 2.58
99.9 3.29
Step 3 :Z is found by
looking Table Value on
Page 811
/ 2α
/ 2α
X
<---CI--
/ 2 / 2X Z X Z
n n
α α
σ σ
µ− ≤ ≤ +
/2αConfidence
Interval

15. © 2003 Prentice-Hall, Inc.
Chap 8-15
Example
A random sample of 10 heights
showed an mean heights of
students in a class was 1.5
meters. It is believed that the
population of Cambodian male
heights is a standard
deviation of 0.1 meters and is
very close to a normal
distribution. Construct a 95%
confidence interval for the
average heights of people in the
class. Interpret your result.
85309 344690µ< <
The 95% CI for the population mean:
Sample mean 1.50
Stand Deviation 0.10
Sample Size (Number of Students) = 15
Confidence Interval 95
Zfrom table P811 1.96
Upper Confidence Interval 1.55
Lower Confidence Interval 1.45
95% Confident Interval that the population
mean is between: -
Microsoft Excel
Worksheet

16. © 2003 Prentice-Hall, Inc.
Chap 8-16
Example: Interpretation
(continued)
We are 95% confident that the Cambodian male number
population average of height is between 1.55 meters
and 1.45 meters.
If all possible samples of size 15 are taken and the
corresponding 95% confidence intervals are constructed,
95% of the confidence intervals that are constructed will
contain the true unknown population mean.
For this particular confidence interval [1.45 meters , 1.55
meters], the unknown population mean can either be in
the interval or not in the interval. It is, therefore,
incorrect to state that the probability is 95% that the
unknown population mean will be in the interval [1.45
meters , 1.55 meters],

17. © 2003 Prentice-Hall, Inc.
Chap 8-17
Example: Interpretation
(continued)
Using the confidence interval method on repeated
sampling, the probability that we will have constructed a
confidence interval that will contain the unknown
population mean is 95%.

18. © 2003 Prentice-Hall, Inc.
Chap 8-18
Factors Affecting Interval Width
(Precision)
 Data Variation
 Measured by
 Sample Size

 Level of Confidence

Intervals Extend from
© 1984-1994 T/Maker Co.
X - Zσ to X + Z σ
xx
σ
X
n
σ
σ =
( )100 1 %α−

19. © 2003 Prentice-Hall, Inc.
Chap 8-19
 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)
µ
σ
/ 2, 1 /2, 1n n
S S
X t X t
n n
α αµ− −− ≤ ≤ +
Margin of Error
Standard Error

20. © 2003 Prentice-Hall, Inc.
Chap 8-20
Student’s t Distribution
Z
t
0
t (df = 5)
t (df = 13)
Bell-Shaped
Symmetric
‘Fatter’
Tails
Standard
Normal

21. © 2003 Prentice-Hall, Inc.
Chap 8-21
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 2.920
t Values
Let: n = 3
df = n - 1 = 2
α = .10
α/2 =.05
α / 2 = .
05

22. © 2003 Prentice-Hall, Inc.
Chap 8-22
Step 3 : The Critical Value t is found by looking Table Value
on Page 812 and Page 813
How to find the t Value based on
a Confidence Interval (CI)
/ 2α
/ 2α
X
X
µ µ=
1 α−
/ 2 X
Zαµ σ+
UTA = / 2α
1, −/2 nα
Step 1. Find / 2α
)
100
CI
(−1=α
Step 2. Find Upper Tail Area (UTA)
t 1, −/2 nα
/ 2, 1 / 2, 1n n
S S
X t X t
n n
α αµ− −− ≤ ≤ +Step 4 : Find CI

23. © 2003 Prentice-Hall, Inc.
Chap 8-23
Example
/2, 1 /2, 1
8 8
50 2.0639 50 2.0639
25 25
46.69 53.30
n n
S S
X t X t
n n
α αµ
µ
µ
− −− ≤ ≤ +
− ≤ ≤ +
≤ ≤
A random sample of 25 has 50 and 8.
Set up a 95% confidence interval estimate for
n X S
µ
= = =
We are 95% confident that the unknown true population
mean is somewhere between 46.69 and 53.30.
.

24. © 2003 Prentice-Hall, Inc.
Chap 8-24
 PHStat | Confidence Interval | Estimate for the
Mean, Sigma Unknown
 Example in Excel Spreadsheet
Confidence Interval for
( Unknown) in PHStat
µ
σ
Microsoft Excel
Worksheet
Confidence Interval Estimate for the Mean
Data
Sample Standard Deviation 8
Sample Mean 50
Sample Size 25
Confidence Level 95%
Standard Error of the Mean 1.6
Degrees of Freedom 24
t Value 2.063898137
Interval Half Width 3.302237019
Interval Lower Limit 46.70
Interval Upper Limit 53.30
Intermediate Calculations
Confidence Interval

25. © 2003 Prentice-Hall, Inc.
Chap 8-25
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 S
S S
p p p p
p Z p p Z
n n
α α
− −
− ≤ ≤ +
Margin of Error
Standard Error

26. © 2003 Prentice-Hall, Inc.
Chap 8-26
Example
A random sample of 400 voters showed that 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 s
s s
p p p p
p Z p p Z
n n
p
p
α α2 2
− −
− ≤ ≤ +
− −
− ≤ ≤ +
≤ ≤
We are 95% confident that the proportion of voters who
prefer Candidate A is somewhere between 0.053 and
0.107.

27. © 2003 Prentice-Hall, Inc.
Chap 8-27
Confidence Interval Estimate for
Proportion in PHStat
 PHStat | Confidence Interval | Estimate for the
Proportion …
 Example in Excel Spreadsheet
Microsoft Excel
Worksheet
Confidence Interval Estimate for the Mean
Data
Sample Size 400
Number of Successes 32
Confidence Level 95%
Sample Proportion 0.08
Z Value -1.95996108
Standard Error of the Proportion 0.01356466
Interval Half Width 0.026586206
Interval Lower Limit 0.053413794
Interval Upper Limit 0.106586206
Intermediate Calculations
Confidence Interval

28. © 2003 Prentice-Hall, Inc.
Chap 8-28
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

29. © 2003 Prentice-Hall, Inc.
Chap 8-29
Chapter Summary
 Illustrated Estimation Process
 Discussed Point Estimates
 Addressed Interval Estimates
 Discussed Confidence Interval Estimation
for the Mean ( Known)
 Discussed Confidence Interval Estimation
for the Mean ( Unknown)
σ
σ

30. © 2003 Prentice-Hall, Inc.
Chap 8-30
Chapter Summary
 Discussed Confidence Interval Estimation for
the Proportion
 Addressed Confidence Interval Estimation
and Ethical Issues
(continued)