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Statistics For Managers
5th Edition
Chapter 8
Confidence Interval
Estimation
Learning Objectives
In this chapter, you learn:
• To construct and interpret confidence
interval estimates for the mean and the
proportion
• How to determine the sample size
necessary to develop a confidence
interval for the mean or proportion
Estimation Process
Population
Mean, , is
unknown
Sample
Random Sample
Mean
X = 50
I am 95%
confident that 
is between 40 &
60.
Point Estimates
We can estimate a
Population Parameter …
with a Sample
Statistic
(a Point Estimate)
Mean
μ
X
Proportion
π
p
Point Estimation
• Assigns a Single Value as the Estimate of
the Parameter
• Attaches a Probabilistic Statement About
the Possible Size of the Error in Doing So
Interval Estimation
• Provides Range of Values
–
Based on Observations from 1 Sample
• Gives Information about Closeness to
Unknown Population Parameter
• Stated in terms of Probability
Never 100% Sure
General Formula
• The general formula for all
confidence intervals is:
Point Estimate ± (Critical Value)(Standard Error)
Elements of Confidence
Interval Estimation
A Probability That the Population Parameter
Falls Somewhere Within the Interval.
Sample
Confidence Interval
Statistic
Confidence Limit
(Lower)
Confidence Limit
(Upper)
Confidence Limits for
Population Mean
Parameter =
Statistic ± Its Error
  X  Error
X  
Z 
= Error =
  X
X  
Error
 X

X

Error  Z 
  X  Z X
© 1984-1994 T/Maker Co.
x
Confidence Intervals
X  Z  X  X  Z 

x_
n
_
X
  1.645  x
  1.645  x
90% Samples
  1.96 x
  1.96 x
95% Samples
  2.58 x
  2.58 x
99% Samples
Level of Confidence
• Probability that the unknown population
parameter falls within the interval
• Denoted (1 - ) % = level of confidence
e.g. 90%, 95%, 99%
 Is Probability That the Parameter Is Not
Within the Interval
Factors Affecting
Interval Width
• Data Variation
•
measured by 
Intervals Extend from
X - Z
x
to X + Z 
x
• Sample Size
X  X / n
• Level of Confidence
(1 - )
© 1984-1994 T/Maker Co.
Confidence Interval Estimates
Confidence
Intervals
Mean
 Known
Proportion
 Unknown
Finite
Population
Confidence Interval for μ
(σ Known)
• Assumptions
– Population standard deviation σ is known
– Population is normally distributed
– If population is not normal, use large sample
• Confidence interval estimate:
σ
XZ
n
Finding the Critical Value, Z
Z  1.96
• Consider a 95% confidence interval:
1   0.95
α
 0.025
2
Z units:
X units:
α
 0.025
2
Z= -1.96
Lower
Confidence
Limit
0
Point Estimate
Z= 1.96
Upper
Confidence
Limit
Common Levels of
Confidence
• Commonly used confidence levels are
90%, 95%, and 99%
Confidence
Level
80%
90%
95%
98%
99%
99.8%
99.9%
Confidence
Coefficient,
Z value
0.80
0.90
0.95
0.98
0.99
0.998
0.999
1.28
1.645
1.96
2.33
2.58
3.08
3.27
1 
Example
• A sample of 11 circuits from a large
normal population has a mean
resistance of 2.20 ohms. We know
from past testing that the population
standard deviation is 0.35 ohms.
• Solution:
σ
X Z
n
 2.20  1.96 (0.35/ 11)
 2.20  0.2068
1.9932    2.4068
Interpretation
• We are 95% confident that the true
mean resistance is between 1.9932
and 2.4068 ohms
• Although the true mean may or may
not be in this interval, 95% of intervals
formed in this manner will contain the
true mean
Confidence Interval for μ
(σ Unknown)
• If the population standard deviation σ is
unknown, we can substitute the sample
standard deviation, S
• This introduces extra uncertainty, since
S is variable from sample to sample
• So we use the t distribution instead of the
normal distribution
Confidence Interval for μ
(σ Unknown)
(continued)
• 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:
X  t n-1
S
n
(where t is the critical value of the t distribution with n -1
degrees of freedom and an area of α/2 in each tail)
Student’s t Distribution
Note: t
Z as n increases
Standard
Normal
(t with df = ∞)
t (df = 13)
t-distributions are bellshaped and symmetric, but
have ‘fatter’ tails than the
normal
t (df = 5)
0
t
Degrees of Freedom (df)
• Number of Observations that Are Free
•
to Vary After Sample Mean Has
Been
degrees of freedom =
•
Calculated
n -1
• Example
= 3 -1
–
Mean of 3 Numbers Is 2
X1 = 1 (or Any Number)
X2 = 2 (or Any Number)
X3 = 3 (Cannot Vary)
Mean = 2
=2
Student’s t Table
/2
Upper Tail Area
df
.25
.10
.05
Assume: n = 3
=n-1=2
df
 = .10
/2 =.05
1 1.000 3.078 6.314
2 0.817 1.886 2.920
.05
3 0.765 1.638 2.353
0
t Values
2.920
t
Example
A random sample of n = 25 taken from a
normal population has X = 50 and S = 8.
Form a 95% confidence interval for μ.
– d.f. = n – 1 = 24, so
t/2 , n1  t 0.025,24  2.0639
The confidence interval is
X  t /2, n-1
S
8
 50  (2.0639)
n
25
46.698 ≤ μ ≤ 53.302
Confidence Interval Estimates
Confidence
Intervals
Mean
 Known
Proportion
 Unknown
Finite
Population
Confidence Intervals for the
Population Proportion, π
(continued)
• Recall that the distribution of the
sample proportion is approximately
normal if the sample size is large, with
standard deviation
σp 
 (1   )
n
p(1  p)
n
• We will estimate this with sample data:
Example
• A random sample of 100 people
shows that 25 are left-handed.
• Form a 95% confidence interval
for the true proportion of lefthanders
Example
(continued)
• A random sample of 100 people
shows that 25 are left-handed. Form
a 95% confidence interval for the true
proportion of left-handers.
p  Z p(1 p)/n
 25/100  1.96 0.25(0.75) /100
 0.25  1.96 (0.0433)
0.1651    0.3349
Sampling Error
• The required sample size needed to estimate a
population parameter to within a selected margin of
error (e) using a specified level of confidence (1 - ) can
be computed
• The margin of error is also called sampling error
– the amount of imprecision in the estimate of the
population parameter
– the amount added and subtracted to the point
estimate to form the confidence interval
Determining Sample Size
(continued)
Determining
Sample Size
For the
Mean
σ
eZ
n
Z σ
n
2
e
2
Now solve
for n to get
2
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.
1.645  45
Z
n

2
2
Error
5
2
2
2
2
  219.2  220
Round Up
Determining Sample Size
• To determine the required sample size
for the proportion, you must know:
– The desired level of confidence (1 - ),
which determines the critical Z value
– The acceptable sampling error, e
– The true proportion of “successes”, π
• π can be estimated with a pilot sample, if
necessary (or conservatively use π = 0.5)
Required Sample Size
Example
How large a sample would be
necessary to estimate the true
proportion defective in a large
population within ±3%, with 95%
confidence?
(Assume a pilot sample yields p =
0.12)
Required Sample Size
Example
(continued)
Solution:
For 95% confidence, use Z = 1.96
e = 0.03
p = 0.12, so use this to estimate π
Z π (1 π ) (1.96) (0.12)(1  0.12)
n

 450.74
2
2
e
(0.03)
2
2
So use n = 451
Estimation for
Finite Populations
• Assumptions
– Sample Is Large Relative to Population
• n / N > .05
• Use Finite Population Correction Factor
• Confidence Interval (Mean, X Unknown)
N n

N 1
N n

N 1
Example: Sample Size
Using the FPC
•What sample size is needed to be 90%
confident of being correct within ± 5?
Suppose the population size N = 500.
n0N
 500
2
.
219
n

 152.6
n0  ( N  1 )
219.2  ( 500  1 )
 153
Round Up
Chapter Summary
•Discussed Confidence Interval Estimation for
the Mean (Known and Unknown)
•Addressed Confidence Interval Estimation for
the Proportion
•Addressed the Situation of Finite Populations
•Determined Sample Size