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Confidence Intervals
(Chapter 8)
• Confidence Intervals for numerical data:
– Standard deviation known
– Standard deviation unknown
• Confidence Intervals for categorical data
Estimation Process: Example
• We are interested in knowing the average
household income  in a certain county.
• A sample with 144 observations yields a sample
mean X=$72,000.
• It is also “known” that in this county, =$24,000
• How can we get a “good” estimate for the true
average household income ? Or:
• How far away (“how bad”) can X be as an
estimate for ?
Estimation Process
Population
Random Sample
Mean, , is
unknown
Mean
X = 50
Sample
I am 95%
confident that
 is between 40
& 60.
Point Estimates
Estimate Population
Parameters …
Mean
Proportion
Variance
Difference

p
with Sample
Statistics
X
PS

1  2
2
S
2
X1  X 2
Interval Estimates
• Provides 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
Confidence Interval Estimates
Confidence
Intervals
Mean
 Known
Proportion
 Unknown
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
X  Z / 2

n
   X  Z / 2

n
General Formula
The general formula for a confidence interval is:
Point Estimate ± Margin of Error
Point Estimate ± (Critical Value)(Standard Error)
Where:
• Point Estimate is the sample statistic estimating the population
parameter of interest
• Critical Value is a table value based on the sampling
distribution of the point estimate and the desired confidence
level
• Standard Error is the standard deviation of the point estimate
Elements of
Confidence Interval Estimation
• Level of confidence
– Confidence in which 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
Level of Confidence
• Denoted by 100 1    %
• A relative frequency interpretation
– In the long run, 100 1    % of all the confidence
intervals that can be constructed will contain the
unknown parameter
• A specific interval will either contain or not
contain the parameter
– No probability involved in a specific interval
Interval and Level of Confidence
Sampling Distribution of the
_ Mean
  Z / 2 X
Intervals
extend from
 /2
X
1
  Z / 2 X
 /2
X  
X  Z X
X
100 1    %
of intervals
constructed
contain  ;
to
X  Z X
100 % do
Confidence Intervals
not.
Factors Affecting
Margin of error (Precision)
• Data variation
– Measured by

• Sample size
–   
X
Intervals Extend from
X - Z
x
to X + Z 
x
n
• Level of confidence
– 100 1    %
© 1984-1994 T/Maker Co.
Determining Sample Size (Cost)
Too Big:
Too small:
• Requires
too much
resources
• Won’t do
the job
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
Do You Ever Truly Know σ?
• Probably not!
• In virtually all real world business situations, σ is not
known.
• If there is a situation where σ is known then µ is also
known (since to calculate σ you need to know µ.)
• If you truly know µ there would be no need to gather a
sample to estimate it.
Confidence Interval for
( Unknown)


• 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 / 2,n 1
S
S
   X  t / 2,n 1
n
n
Student’s t Distribution
Standard
Normal
Bell-Shaped
Symmetric
‘Fatter’
Tails
t (df = 13)
t (df = 5)
0
Z
t
Example
A random sample of n  25 has X  50 and S  8.
Set up a 95% confidence interval estimate for 
S
S
X  t / 2,n 1
   X  t / 2,n 1
n
n
8
8
50  2.0639
   50  2.0639
25
25
46.69    53.30
Confidence Interval
Estimate for Proportion
• Assumptions
– Two categorical outcomes
– Population follows binomial distribution
– Normal approximation can be used if
np  5 and n 1  p  5
– Confidence interval estimate
–
pS 1  pS 
pS 1  pS 
pS  Z / 2
 p  pS  Z / 2
n
n


Example
A random sample of 400 Voters showed 32 preferred
Candidate A. Set up a 95% confidence interval
estimate for p.
ps  Z / 
ps 1  ps 
 p  ps  Z / 
n
ps 1  ps 
n
.08 1  .08 
.08 1  .08 
.08  1.96
 p  .08  1.96
400
400
.053  p  .107
Determining Sample Size
for Proportion
Out of a population of 1,000, we randomly
selected 100 of which 30 were defective. What
sample size is needed to be within ± 5% with 90%
confidence?
Z p 1  p  1.645  0.3 0.7 
n

2
2
Error
0.05
 227.3  228
2
2
Round Up
Excel Tutorial
Constructing Confidence Intervals using Excel:
• Tutorial
•Excel spreadsheet