Transcript Slide 1
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 = Error = X
Z
X
X
Error
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
XZ
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, n1 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
eZ
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)
Nn
N1
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