Download Confidence Interval Estimates

© 2000 Prentice-Hall, Inc.
Statistics for Business and
Economics
Inferences Based on a Single Sample:
Estimation with Confidence Intervals
Chapter 7
7-1
Learning Objectives
© 2000 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
© 2000 Prentice-Hall, Inc.
Suppose you’re
interested in the
average amount of
money that students
in this class (the
population) have on
them. How would
you find out?
7-3
© 2000 Prentice-Hall, Inc.
Introduction
to Estimation
7-4
Statistical Methods
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Statistical
Methods
Descriptive
Statistics
Inferential
Statistics
Estimation
7-5
Hypothesis
Testing
Estimation Process
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7-6
Estimation Process
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Population
Mean, , is
unknown

 
 


 
7-7
Estimation Process
© 2000 Prentice-Hall, Inc.
Population


Mean, , is
unknown

 
Sample


 
7-8
Random Sample
Mean 
X = 50
Estimation Process
© 2000 Prentice-Hall, Inc.
Population


Mean, , is
unknown

 
Sample


 
7-9
Random Sample
Mean 
X = 50
I am 95%
confident that
 is between
40 & 60.
© 2000 Prentice-Hall, Inc.
Unknown Population
Parameters Are Estimated
Estimate Population
Parameter...
Mean

Proportion
p
Variance

Differences
7 - 10
2
1 -  2
with Sample
Statistic
x
p^
s
2
x1 -x2
Estimation Methods
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7 - 11
Estimation Methods
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Estimation
7 - 12
Estimation Methods
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Estimation
Point
Estimation
7 - 13
Estimation Methods
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Estimation
Point
Estimation
7 - 14
Interval
Estimation
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Point Estimation
7 - 15
Estimation Methods
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Estimation
Point
Estimation
7 - 16
Interval
Estimation
Point Estimation
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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 - 17
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Interval Estimation
7 - 18
Estimation Methods
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Estimation
Point
Estimation
7 - 19
Interval
Estimation
Interval Estimation
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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 - 20
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7 - 21
Key Elements of
Interval Estimation
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Key Elements of
Interval Estimation
Sample statistic
(point estimate)
7 - 22
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Key Elements of
Interval Estimation
Confidence
interval
Confidence
limit (lower)
7 - 23
Sample statistic
(point estimate)
Confidence
limit (upper)
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Key Elements of
Interval Estimation
A probability that the population parameter
falls somewhere within the interval.
Confidence
interval
Confidence
limit (lower)
7 - 24
Sample statistic
(point estimate)
Confidence
limit (upper)
Confidence Limits
for Population Mean
© 2000 Prentice-Hall, Inc.
Parameter =
Statistic ± Error
© 1984-1994
T/Maker Co.
7 - 25
(1)
  X  Error
(2)
Error  X   or X  
X 
(3)
Z
(4)
Error  Z x
(5)
  X  Z x
x

Error
x
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7 - 26
Many Samples Have
Same Interval
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Many Samples Have
Same Interval
x_

7 - 27
X
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Many Samples Have
Same Interval
X =  ± Zx
x_

7 - 28
X
© 2000 Prentice-Hall, Inc.
Many Samples Have
Same Interval
X =  ± Zx
x_
-1.65x

+1.65x
90% Samples
7 - 29
X
© 2000 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 - 30
X
© 2000 Prentice-Hall, Inc.
Many Samples Have
Same Interval
X =  ± Zx
x_
-2.58x
-1.65x
-1.96x

+1.65x
+2.58x
+1.96x
90% Samples
95% Samples
99% Samples
7 - 31
X
Confidence Level
© 2000 Prentice-Hall, Inc.
1. Probability that the Unknown
Population Parameter Falls Within
Interval
2. Denoted (1 - 

 Is Probability That Parameter Is Not
Within Interval
3. Typical Values Are 99%, 95%, 90%
7 - 32
© 2000 Prentice-Hall, Inc.
Intervals &
Confidence Level
Sampling
Distribution /2
of Mean
x_
1 -
/2
x = 
X
(1 - ) % of
intervals
contain .
Intervals
extend from
X - ZX to
X + ZX
 % do not.
Large number of intervals
7 - 33
_
Factors Affecting
Interval Width
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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 - 34
© 2000 Prentice-Hall, Inc.
7 - 35
Confidence Interval
Estimates
© 2000 Prentice-Hall, Inc.
Confidence Interval
Estimates
Confidence
Intervals
7 - 36
© 2000 Prentice-Hall, Inc.
Confidence Interval
Estimates
Confidence
Intervals
Mean
7 - 37
© 2000 Prentice-Hall, Inc.
Confidence Interval
Estimates
Confidence
Intervals
Mean
7 - 38
Proportion
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Confidence Interval
Estimates
Confidence
Intervals
Mean
7 - 39
Proportion
Variance
Confidence Interval
Estimates
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Confidence
Intervals
Mean
Known
7 - 40
Proportion
Variance
Confidence Interval
Estimates
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Confidence
Intervals
Mean
Known
7 - 41
Proportion
 Unknown
Variance
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Confidence Interval Estimate
Mean ( Known)
7 - 42
Confidence Interval
Estimates
© 2000 Prentice-Hall, Inc.
Confidence
Intervals
Mean
Known
7 - 43
Proportion
 Unknown
Variance
Confidence Interval
Mean ( Known)
© 2000 Prentice-Hall, Inc.
1. Assumptions



Population Standard Deviation Is Known
Population Is Normally Distributed
If Not Normal, Can Be Approximated by
Normal Distribution (n  30)
7 - 44
Confidence Interval
Mean ( Known)
© 2000 Prentice-Hall, Inc.

1.Assumptions



Population Standard Deviation Is Known
Population Is Normally Distributed
If Not Normal, Can Be Approximated by
Normal Distribution (n  30)
2. Confidence Interval Estimate
X  Z / 2 
7 - 45

n
   X  Z / 2 

n
© 2000 Prentice-Hall, Inc.
Estimation Example
Mean ( Known)
The mean of a random sample of n = 25
isX = 50. Set up a 95% confidence
interval estimate for  if  = 10.
7 - 46
© 2000 Prentice-Hall, Inc.
Estimation Example
Mean ( Known)
The mean of a random sample of n = 25
isX = 50. Set up a 95% confidence
interval estimate for  if  = 10.


X  Z / 2 
   X  Z / 2 
n
n
10
10
50  1.96 
   50  1.96 
25
25
46.08    53.92
7 - 47
Thinking Challenge
© 2000 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 - 48
© 2000 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 - 49
© 2000 Prentice-Hall, Inc.
Confidence Interval Estimate
Mean ( Unknown)
7 - 50
Confidence Interval
Estimates
© 2000 Prentice-Hall, Inc.
Confidence
Intervals
Mean
Known
7 - 51
Proportion
 Unknown
Variance
Confidence Interval
Mean ( Unknown)
© 2000 Prentice-Hall, Inc.
1. Assumptions


Population Standard Deviation Is Unknown
Population Must Be Normally Distributed
2. Use Student’s t Distribution
7 - 52
Confidence Interval
Mean ( Unknown)
© 2000 Prentice-Hall, Inc.
1. Assumptions


Population Standard Deviation Is Unknown
Population Must Be Normally Distributed
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 - 53
Student’s t Distribution
© 2000 Prentice-Hall, Inc.
Standard
Normal
Bell-Shaped
t (df = 13)
Symmetric
t (df = 5)
‘Fatter’ Tails
0
7 - 54
Z
t
Student’s t Table
© 2000 Prentice-Hall, Inc.
7 - 55
Student’s t Table
© 2000 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 - 56
Student’s t Table
© 2000 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 - 57
Student’s t Table
© 2000 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 - 58
t
Student’s t Table
© 2000 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 - 59
t
Student’s t Table
© 2000 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 - 60
t
Student’s t Table
© 2000 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 - 61
t
Student’s t Table
© 2000 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 - 62
2.920
t
Degrees of Freedom (df)
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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 - 63
degrees of freedom
= n -1
= 3 -1
=2
© 2000 Prentice-Hall, Inc.
Estimation Example
Mean ( Unknown)
A random sample of n = 25 hasx = 50 & s
= 8. Set up a 95% confidence interval
estimate for .
7 - 64
© 2000 Prentice-Hall, Inc.
Estimation Example
Mean ( Unknown)
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 - 65
S
n
8
25
Thinking Challenge
© 2000 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 - 66
© 2000 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 - 67
© 2000 Prentice-Hall, Inc.
Confidence Interval Estimate
of Proportion
7 - 68
Confidence Interval
Estimates
© 2000 Prentice-Hall, Inc.
Confidence
Intervals
Mean
Known
7 - 69
Proportion
 Unknown
Variance
© 2000 Prentice-Hall, Inc.
7 - 70
Confidence Interval
Proportion
Confidence Interval
Proportion
© 2000 Prentice-Hall, Inc.
1. Assumptions



Two Categorical Outcomes
Population Follows Binomial Distribution
Normal Approximation Can Be Used

7 - 71
npˆ  3 npˆ 1  pˆ  Does Not Include 0 or 1
Confidence Interval
Proportion
© 2000 Prentice-Hall, Inc.
1. Assumptions



Two Categorical Outcomes
Population Follows Binomial Distribution
Normal Approximation Can Be Used

npˆ  3 npˆ 1  pˆ  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
7 - 72
© 2000 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.
7 - 73
© 2000 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
7 - 74
Thinking Challenge
© 2000 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?
7 - 75
© 2000 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
7 - 76
© 2000 Prentice-Hall, Inc.
Finding Sample Sizes
7 - 77
Finding Sample Sizes
for Estimating 
© 2000 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
7 - 78
I don’t want to
sample too much
or too little!

n
Sample Size Example
© 2000 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.
7 - 79
Sample Size Example
© 2000 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.

Z
1.645 45
n

 219.2  220
2
2
Error
5
2
7 - 80
2
2
2
Thinking Challenge
© 2000 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?
7 - 81
Sample Size Solution*
© 2000 Prentice-Hall, Inc.
Z 2 2
n
Error 2
2
2

1.96  400 

502
 245.86  246
7 - 82
Conclusion
© 2000 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
7 - 83
End of Chapter
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