Download Confidence Interval Estimates

© 1998 Prentice-Hall, Inc.
Chapter 7
Inferences Based on a Single Sample:
Estimation with Confidence Intervals
7-1
Learning Objectives
© 1998 Prentice-Hall, Inc.
1. State what is estimated
2. Distinguish point & interval estimates
3. Explain interval estimates
4. Compute confidence interval estimates
for population mean & proportion
5. Compute sample size
7-2
Thinking Challenge
© 1998 Prentice-Hall, Inc.
Suppose you’re
interested in estimating
the average amount of
money that second-year
business students
(population) have on
them. How would you
find out?
7-3
Alone
Group Class
© 1998 Prentice-Hall, Inc.
Introduction
to Estimation
7-4
© 1998 Prentice-Hall, Inc.
Types of
Statistical Applications
Statistical
Methods
Descriptive
Statistics
Inferential
Statistics
Estimation
7-5
Hypothesis
Testing
Estimation Process
© 1998 Prentice-Hall, Inc.
7-6
Estimation Process
© 1998 Prentice-Hall, Inc.
Population
Mean, , is
unknown

 
 


 
7-7
Estimation Process
© 1998 Prentice-Hall, Inc.
Population


Mean, , is
unknown

 
Sample


 
7-8
Random Sample
Mean 
X = 50
Estimation Process
© 1998 Prentice-Hall, Inc.
Population


Mean, , is
unknown

 
Sample


 
7-9
Random Sample
Mean 
X = 50
I am 95%
confident that
 is between
42 & 58.
© 1998 Prentice-Hall, Inc.
Unknown Population
Parameters Are Estimated
7 - 10
© 1998 Prentice-Hall, Inc.
Unknown Population
Parameters Are Estimated
Estimate population
parameter...
7 - 11
with sample
statistic
© 1998 Prentice-Hall, Inc.
Unknown Population
Parameters Are Estimated
Estimate population
parameter...
Mean

7 - 12
with sample
statistic
x
© 1998 Prentice-Hall, Inc.
Unknown Population
Parameters Are Estimated
Estimate population
parameter...
Mean

Proportion
7 - 13
p
with sample
statistic
x
p^
© 1998 Prentice-Hall, Inc.
Unknown Population
Parameters Are Estimated
Estimate population
parameter...
Mean

Proportion
p
Variance

7 - 14
2
with sample
statistic
x
p^
s
2
© 1998 Prentice-Hall, Inc.
Unknown Population
Parameters Are Estimated
Estimate population
parameter...
Mean

Proportion
p
Variance

Differences
7 - 15
2
1 -  2
with sample
statistic
x
p^
s
2
x1 -x2
Estimation Methods
© 1998 Prentice-Hall, Inc.
7 - 16
Estimation Methods
© 1998 Prentice-Hall, Inc.
Estimation
7 - 17
Estimation Methods
© 1998 Prentice-Hall, Inc.
Estimation
Point
Estimation
7 - 18
Estimation Methods
© 1998 Prentice-Hall, Inc.
Estimation
Point
Estimation
7 - 19
Interval
Estimation
© 1998 Prentice-Hall, Inc.
Point Estimation
7 - 20
Estimation Methods
© 1998 Prentice-Hall, Inc.
Estimation
Point
Estimation
7 - 21
Interval
Estimation
Point Estimation
© 1998 Prentice-Hall, Inc.
1. Provides single value

Based on observations from 1 sample
2. Gives no information about how close
value is to the unknown population
parameter
3. Example: Sample meanx = 3 is point
estimate of unknown population mean
7 - 22
© 1998 Prentice-Hall, Inc.
Interval Estimation
7 - 23
Estimation Methods
© 1998 Prentice-Hall, Inc.
Estimation
Point
Estimation
7 - 24
Interval
Estimation
Interval Estimation
© 1998 Prentice-Hall, Inc.
1. Provides range of values

Based on observations from 1 sample
2. Gives information about closeness to
unknown population parameter

Stated in terms of probability
 Knowing exact closeness requires knowing
unknown population parameter
3. Example: unknown population mean lies
between 50 & 70 with 95% confidence
7 - 25
© 1998 Prentice-Hall, Inc.
7 - 26
Key Elements of
Interval Estimation
© 1998 Prentice-Hall, Inc.
Key Elements of
Interval Estimation
Sample statistic
(point estimate)
7 - 27
© 1998 Prentice-Hall, Inc.
Key Elements of
Interval Estimation
Confidence
interval
7 - 28
Sample statistic
(point estimate)
© 1998 Prentice-Hall, Inc.
Key Elements of
Interval Estimation
Confidence
interval
Confidence
limit (lower)
7 - 29
Sample statistic
(point estimate)
Confidence
limit (upper)
© 1998 Prentice-Hall, Inc.
Key Elements of
Interval Estimation
A probability that the population parameter
falls somewhere within the interval.
Confidence
interval
Confidence
limit (lower)
7 - 30
Sample statistic
(point estimate)
Confidence
limit (upper)
Confidence Limits
for Population Mean
© 1998 Prentice-Hall, Inc.
Parameter =
Statistic ± Error
© 1984-1994
T/Maker Co.
7 - 31
(1)
  X  Error
(2)
Error  X   or X  
X 
(3)
Z
(4)
Error  Z x
(5)
  X  Z x
x

Error
x
© 1998 Prentice-Hall, Inc.
7 - 32
Many Samples Have
Same Interval
© 1998 Prentice-Hall, Inc.
Many Samples Have
Same Interval
x_

7 - 33
X
© 1998 Prentice-Hall, Inc.
Many Samples Have
Same Interval
X =  ± Zx
x_

7 - 34
X
© 1998 Prentice-Hall, Inc.
Many Samples Have
Same Interval
X =  ± Zx
x_
-1.65x

+1.65x
90% Samples
7 - 35
X
© 1998 Prentice-Hall, Inc.
Many Samples Have
Same Interval
X =  ± Zx
x_
-1.65x
-1.96x

+1.65x
+1.96x
90% Samples
95% Samples
7 - 36
X
© 1998 Prentice-Hall, Inc.
Many Samples Have
Same Interval
X =  ± Zx
x_
-2.58x
-1.645x
-1.96x

+1.645x +2.58x
+1.96x
90% Samples
95% Samples
99% Samples
7 - 37
X
Confidence Level
© 1998 Prentice-Hall, Inc.
1. Probability that the unknown population
parameter falls within interval
2. Denoted (1 - 100

 is probability that parameter is not
within interval
3. Typical values are 99%, 95%, 90%
7 - 38
© 1998 Prentice-Hall, Inc.
7 - 39
Intervals &
Confidence Level
© 1998 Prentice-Hall, Inc.
Intervals &
Confidence Level
Sampling
Distribution /2
of Mean
x_
1 -
x = 
7 - 40
/2
_
X
© 1998 Prentice-Hall, Inc.
Intervals &
Confidence Level
Sampling
Distribution /2
of Mean
x_
1 -
x = 
/2
_
X
Confidence
Interval
7 - 41
© 1998 Prentice-Hall, Inc.
Intervals &
Confidence Level
Sampling
Distribution /2
of Mean
x_
1 -
x = 
/2
_
X
Confidence
Interval
7 - 42
© 1998 Prentice-Hall, Inc.
Intervals &
Confidence Level
Sampling
Distribution /2
of Mean
x_
1 -
x = 
Intervals
extend from
X - ZX to
X + ZX
7 - 43
/2
_
X
Confidence
Interval
© 1998 Prentice-Hall, Inc.
Intervals &
Confidence Level
Sampling
Distribution /2
of Mean
x_
1 -
x = 
Intervals
extend from
X - ZX to
X + ZX
7 - 44
/2
_
X
Confidence
Interval
© 1998 Prentice-Hall, Inc.
Intervals &
Confidence Level
Sampling
Distribution /2
of Mean
x_
1 -
x = 
Intervals
extend from
X - ZX to
X + ZX
7 - 45
/2
_
X
Confidence
Interval
© 1998 Prentice-Hall, Inc.
Intervals &
Confidence Level
Sampling
Distribution /2
of Mean
x_
1 -
/2
x = 
Intervals
extend from
X - ZX to
X + ZX
X
Confidence
Interval
Large number of intervals
7 - 46
_
© 1998 Prentice-Hall, Inc.
Intervals &
Confidence Level
Sampling
Distribution /2
of Mean
x_
1 -
x = 
Intervals
extend from
X - ZX to
X + ZX
7 - 47
/2
_
X
(1 - )100 %
of intervals
contain .
100 % do
Large number of intervals not.
Factors Affecting
Interval Width
© 1998 Prentice-Hall, Inc.
1. Data dispersion

Measured by 
Intervals extend from
X - ZX toX + ZX
2. Sample size

X =  / n
3. Level of confidence
(1 - )

Affects Z
© 1984-1994 T/Maker Co.
7 - 48
© 1998 Prentice-Hall, Inc.
7 - 49
Confidence Interval
Estimates
© 1998 Prentice-Hall, Inc.
Confidence Interval
Estimates
One
Population
7 - 50
© 1998 Prentice-Hall, Inc.
Confidence Interval
Estimates
One
Population
Mean
7 - 51
© 1998 Prentice-Hall, Inc.
Confidence Interval
Estimates
One
Population
Mean
7 - 52
Proportion
© 1998 Prentice-Hall, Inc.
Confidence Interval
Estimates
One
Population
Large
Sample
Mean
Z
Distribution
7 - 53
Proportion
© 1998 Prentice-Hall, Inc.
Confidence Interval
Estimates
One
Population
Large
Sample
Mean
Z
Distribution
7 - 54
Small
Sample
t
Distribution
Proportion
© 1998 Prentice-Hall, Inc.
Confidence Interval
Estimates
One
Population
Large
Sample
Mean
Z
Distribution
7 - 55
Small
Sample
t
Distribution
Proportion
Z
Distribution
© 1998 Prentice-Hall, Inc.
Confidence Interval Estimate
Mean (Large Sample)
7 - 56
© 1998 Prentice-Hall, Inc.
Confidence Interval
Estimates
One
Population
Large
Sample
Mean
Z
Distribution
7 - 57
Small
Sample
t
Distribution
Proportion
Z
Distribution
Confidence Interval
Mean (Large Sample)
© 1998 Prentice-Hall, Inc.
1. Assumptions



Sample size at least 30 (n  30)
Random sample drawn
If population standard deviation unknown,
use sample standard deviation
7 - 58
Confidence Interval
Mean (Large Sample)
© 1998 Prentice-Hall, Inc.
1. Assumptions



Sample size at least 30 (n  30)
Random sample drawn
If population standard deviation unknown,
use sample standard deviation
2. Confidence interval estimate
X  Z / 2 
7 - 59

n
   X  Z / 2 

n
© 1998 Prentice-Hall, Inc.
Estimation Example
Mean (Large Sample)
The mean of a random sample of n = 36
isX = 50. Set up a 95% confidence
interval estimate for  if  = 12.
7 - 60
© 1998 Prentice-Hall, Inc.
Estimation Example
Mean (Large Sample)
The mean of a random sample of n = 36
isX = 50. Set up a 95% confidence
interval estimate for  if  = 12.
X  Z / 2 

   X  Z / 2 

n
n
12
12
50  1.96 
   50  1.96 
36
36
46.08    53.92
7 - 61
Thinking Challenge
© 1998 Prentice-Hall, Inc.
You’re a Q/C inspector for
Gallo. The  for 2-liter
bottles is .05 liters. A
random sample of 100
bottles showedX = 1.99
liters. What is the 90%
confidence interval
estimate of the true mean
amount in 2-liter bottles?
22
liter
liter
© 1984-1994 T/Maker Co.
7 - 62
Alone
Group Class
© 1998 Prentice-Hall, Inc.
Confidence Interval
Solution*
X  Z / 2 

n
   X  Z / 2 

n
.05
.05
1.99  1.645 
   1.99  1.645 
100
100
1.982    1.998
7 - 63
© 1998 Prentice-Hall, Inc.
Confidence Interval Estimate
Mean (Small Sample)
7 - 64
© 1998 Prentice-Hall, Inc.
Confidence Interval
Estimates
One
Population
Large
Sample
Mean
Z
Distribution
7 - 65
Small
Sample
t
Distribution
Proportion
Z
Distribution
Confidence Interval
Mean (Small Sample)
© 1998 Prentice-Hall, Inc.
1. Assumptions



Sample size less than 30 (n < 30)
Population normally distributed
Population standard deviation unknown
2. Use Student’s t distribution
7 - 66
Confidence Interval
Mean (Small Sample)
© 1998 Prentice-Hall, Inc.
1. Assumptions



Sample size less than 30 (n < 30)
Population normally distributed
Population standard deviation unknown
2. Use Student’s t distribution
3. Confidence interval estimate
S
S
X  t  / 2, n 1 
   X  t  / 2, n 1 
n
n
7 - 67
Student’s t Distribution
© 1998 Prentice-Hall, Inc.
7 - 68
Student’s t Distribution
© 1998 Prentice-Hall, Inc.
Standard
Normal
Z
0
7 - 69
Student’s t Distribution
© 1998 Prentice-Hall, Inc.
Standard
Normal
Bell-Shaped
t (df = 13)
Symmetric
‘Fatter’ Tails
0
7 - 70
Z
t
Student’s t Distribution
© 1998 Prentice-Hall, Inc.
Standard
Normal
Bell-Shaped
t (df = 13)
Symmetric
t (df = 5)
‘Fatter’ Tails
0
7 - 71
Z
t
Student’s t Table
© 1998 Prentice-Hall, Inc.
7 - 72
Student’s t Table
© 1998 Prentice-Hall, Inc.
v
t.10
t.05
t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
7 - 73
Student’s t Table
© 1998 Prentice-Hall, Inc.
v
t.10
t.05
t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
t values
7 - 74
Student’s t Table
© 1998 Prentice-Hall, Inc.
/2
v
t.10
t.05
t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
/2
0
t values
7 - 75
t
Student’s t Table
© 1998 Prentice-Hall, Inc.
Assume:
n=3
df = n - 1 = 2
 = .10
/2 =.05
/2
v
t.10
t.05
t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
/2
0
t values
7 - 76
t
Student’s t Table
© 1998 Prentice-Hall, Inc.
Assume:
n=3
df = n - 1 = 2
 = .10
/2 =.05
/2
v
t.10
t.05
t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
/2
0
t values
7 - 77
t
Student’s t Table
© 1998 Prentice-Hall, Inc.
Assume:
n=3
df = n - 1 = 2
 = .10
/2 =.05
/2
v
t.10
t.05
t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
.05
0
t values
7 - 78
t
Student’s t Table
© 1998 Prentice-Hall, Inc.
Assume:
n=3
df = n - 1 = 2
 = .10
/2 =.05
/2
v
t.10
t.05
t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
.05
0
t values
7 - 79
2.920
t
Degrees of Freedom (df)
© 1998 Prentice-Hall, Inc.
1. Number of observations that are free to
vary after sample statistic has been
calculated
7 - 80
Degrees of Freedom (df)
© 1998 Prentice-Hall, Inc.
1. Number of observations that are free to
vary after sample statistic has been
calculated
2. Example:
Sum of 3 numbers is 6
X1 =
X2 =
X3 =
Sum = 6
7 - 81
Degrees of Freedom (df)
© 1998 Prentice-Hall, Inc.
1. Number of observations that are free to
vary after sample statistic has been
calculated
2. Example:
Sum of 3 numbers is 6
X1 = 1 (Or any number)
X2 =
X3 =
Sum = 6
7 - 82
Degrees of Freedom (df)
© 1998 Prentice-Hall, Inc.
1. Number of observations that are free to
vary after sample statistic has been
calculated
2. Example:
Sum of 3 numbers is 6
X1 = 1 (Or any number)
X2 = 2 (Or any number)
X3 =
Sum = 6
7 - 83
Degrees of Freedom (df)
© 1998 Prentice-Hall, Inc.
1. Number of observations that are free to
vary after sample statistic has been
calculated
2. Example:
Sum of 3 numbers is 6
X1 = 1 (Or any number)
X2 = 2 (Or any number)
X3 = 3 (Cannot vary)
Sum = 6
7 - 84
Degrees of Freedom (df)
© 1998 Prentice-Hall, Inc.
1. Number of observations that are free to
vary after sample statistic has been
calculated
2. Example:
Sum of 3 numbers is 6
X1 = 1 (Or any number)
X2 = 2 (Or any number)
X3 = 3 (Cannot vary)
Sum = 6
7 - 85
degrees of freedom
= n -1
= 3 -1
=2
© 1998 Prentice-Hall, Inc.
Estimation Example
Mean (Small Sample)
A random sample of n = 25 hasx = 50 & s
= 8. Set up a 95% confidence interval
estimate for .
7 - 86
© 1998 Prentice-Hall, Inc.
Estimation Example
Mean (Small Sample)
A random sample of n = 25 hasx = 50 & s
= 8. Set up a 95% confidence interval
estimate for .
S
X  t  / 2, n 1 
   X  t  / 2, n 1 
n
8
50  2.0639 
   50  2.0639 
25
46.69    53.30
7 - 87
S
n
8
25
Thinking Challenge
© 1998 Prentice-Hall, Inc.
You’re a time study
analyst in manufacturing.
You’ve recorded the
following task times (min.):
3.6, 4.2, 4.0, 3.5, 3.8, 3.1.
What is the 90%
confidence interval
estimate of the population
mean task time?
7 - 88
Alone
Group Class
© 1998 Prentice-Hall, Inc.
Confidence Interval
Solution*
X = 3.7
S = 3.8987
n = 6, df = n - 1 = 6 - 1 = 5
S / n = 3.8987 / 6 = 1.592
t.05,5 = 2.0150
3.7 - (2.015)(1.592) 3.7 + (2.015)(1.592)
0.492  6.908
7 - 89
© 1998 Prentice-Hall, Inc.
Confidence Interval Estimate
of Proportion
7 - 90
Data Types
© 1998 Prentice-Hall, Inc.
Data
Quantitative
Discrete
7 - 91
Continuous
Qualitative
Qualitative Data
© 1998 Prentice-Hall, Inc.
1. Qualitative random variables yield
responses that classify

e.g., gender (male, female)
2. Measurement reflects # in category
3. Nominal or ordinal scale
4. Examples


Do you own savings bonds?
Do you live on-campus or off-campus?
7 - 92
Proportions
© 1998 Prentice-Hall, Inc.
1. Involve qualitative variables
2. Fraction or % of population in a category
3. If two qualitative outcomes, binomial
distribution

Possess or don’t possess characteristic
7 - 93
Proportions
© 1998 Prentice-Hall, Inc.
1. Involve qualitative variables
2. Fraction or % of population in a category
3. If two qualitative outcomes, binomial
distribution
Possess or don’t possess characteristic
^
4. Sample proportion (p)

x number of successes

p 
n
sample size
7 - 94
© 1998 Prentice-Hall, Inc.
7 - 95
Sampling Distribution
of Proportion
© 1998 Prentice-Hall, Inc.
Sampling Distribution
of Proportion
Sampling Distribution
^
P(P )
.3
.2
.1
.0
^
P
.0
7 - 96
.2
.4
.6
.8
1.0
Sampling Distribution
of Proportion
© 1998 Prentice-Hall, Inc.
1.
Approximated by
normal distribution
b g
np  3 np 1  p
excludes 0 or n
Sampling Distribution
^
P(P )
.3
.2
.1
.0
^
P
.0
7 - 97
.2
.4
.6
.8
1.0
Sampling Distribution
of Proportion
© 1998 Prentice-Hall, Inc.
1.
Approximated by
normal distribution
b g
np  3 np 1  p
excludes 0 or n
2.
Mean
 P  p
7 - 98
Sampling Distribution
^
P(P )
.3
.2
.1
.0
^
P
.0
.2
.4
.6
 P  p
.8
1.0
Sampling Distribution
of Proportion
© 1998 Prentice-Hall, Inc.
1.
Approximated by
normal distribution
b g
np  3 np 1  p
excludes 0 or n
2.
3.
Mean
 P  p
^
P(P )
.3
.2
.1
.0
^
P
.0
Standard error
p  1 p
 p^ 
n
a
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Sampling Distribution
f
.2
.4
.6
 P  p
.8
1.0
© 1998 Prentice-Hall, Inc.
Confidence Interval
Estimates
One
Population
Large
Sample
Mean
Z
Distribution
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Small
Sample
t
Distribution
Proportion
Z
Distribution
© 1998 Prentice-Hall, Inc.
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Confidence Interval
Proportion
Confidence Interval
Proportion
© 1998 Prentice-Hall, Inc.
1. Assumptions



Two categorical outcomes
Population follows binomial distribution
Normal approximation can be used

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b g
np  3 np 1  p does not include 0 or 1
Confidence Interval
Proportion
© 1998 Prentice-Hall, Inc.
1. Assumptions



Two categorical outcomes
Population follows binomial distribution
Normal approximation can be used

b g
np  3 np 1  p does not include 0 or 1
2. Confidence interval estimate
  (1  p )
  (1  p )
p
p
p  z 2 
 p  p  z 2 
n
n
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© 1998 Prentice-Hall, Inc.
Estimation Example
Proportion
A random sample of 400 graduates
showed 32 went to grad school. Set up a
95% confidence interval estimate for p.
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© 1998 Prentice-Hall, Inc.
Estimation Example
Proportion
A random sample of 400 graduates
showed 32 went to grad school. Set up a
95% confidence interval estimate for p.
  (1  p )
  (1  p )
p
p
p  Z  / 2 
 p  p  Z  / 2 
n
n
.08  (1 .08)
.08  (1 .08)
.08  1.96 
 p  .08  1.96 
400
400
.053  p  .107
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Thinking Challenge
© 1998 Prentice-Hall, Inc.
You’re a production
manager for a newspaper.
You want to find the %
defective. Of 200
newspapers, 35 had
defects. What is the
90% confidence interval
estimate of the population
proportion defective?
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Alone
Group Class
© 1998 Prentice-Hall, Inc.
Confidence Interval
Solution*
  (1  p )
  (1  p )
p
p
p  z / 2 
 p  p  z / 2 
n
n
.175  (.825)
.175  (.825)
.175  1.645 
 p  .175  1.645 
200
200
.1308  p  .2192
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© 1998 Prentice-Hall, Inc.
Finding Sample Sizes
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© 1998 Prentice-Hall, Inc.
7 - 109
Finding Sample Sizes
for Estimating 
Finding Sample Sizes
for Estimating 
© 1998 Prentice-Hall, Inc.
(1)
Z
X 
x

Error
x
(2)
Error  Z x  Z
(3)
Z 
n
Error 2
2
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2

n
Finding Sample Sizes
for Estimating 
© 1998 Prentice-Hall, Inc.
(1)
Z
X 
x

Error
x
(2)
Error  Z x  Z
(3)
Z 
n
Error 2
2
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2

n
Finding Sample Sizes
for Estimating 
© 1998 Prentice-Hall, Inc.
(1)
Z
X 
x

Error
x
(2)
Error  Z x  Z
(3)
Z 
n
Error 2
2
2
Error is also called bound, B
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
n
Finding Sample Sizes
for Estimating 
© 1998 Prentice-Hall, Inc.
(1)
Z
X 
x

Error
x
(2)
Error  Z x  Z
(3)
Z 
n
Error 2
2
2
Error is also called bound, B
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I don’t want to
sample too much
or too little!

n
Sample Size Example
© 1998 Prentice-Hall, Inc.
What sample size is needed to be 90%
confident of being correct within  5? A
pilot study suggested that the standard
deviation is 45.
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Sample Size Example
© 1998 Prentice-Hall, Inc.
What sample size is needed to be 90%
confident of being correct within  5? A
pilot study suggested that the standard
deviation is 45.
a f a f  219.2  220
af
2
1645
.
45
Z 
n

2
2
Error
5
2
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2
2
Thinking Challenge
© 1998 Prentice-Hall, Inc.
You work in Human
Resources at Merrill Lynch.
You plan to survey employees
to find their average medical
expenses. You want to be
95% confident that the
sample mean is within ± $50.
A pilot study showed that 
was about $400. What
sample size do you use?
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Alone
Group Class
Sample Size Solution*
© 1998 Prentice-Hall, Inc.
Z 2 2
n
2
Error
400f
1.96f a
a

a50f
2
2
2
 245.86  246
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Conclusion
© 1998 Prentice-Hall, Inc.
1. Stated what is estimated
2. Distinguished point & interval estimates
3. Explained interval estimates
4. Computed confidence interval estimates
for population mean & proportion
5. Computed sample size
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This Class...
© 1998 Prentice-Hall, Inc.
Please take a moment to answer the
following questions in writing:
1. What was the most important thing you
learned in class today?
2. What do you still have questions about?
3. How can today’s class be improved?
7 - 119
End of Chapter
Any blank slides that follow are
blank intentionally.