Module 5
Confidence Interval Estimation
Learning Objectives
• Estimation process
o Point estimates
o Interval estimates
• Confidence interval estimation for the
o Population Mean, μ
• when Population Standard Deviation σ is Known
• when Population Standard Deviation σ is Unknown
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Learning Objectives 2
• Determining Sample Size, n
o when estimating the Population Mean, μ
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Point and Interval Estimates
•
A point estimate is the value of a single sample statistic.
•
A confidence interval provides a range of values
constructed around the point estimate.
Lower
Confidence
Limit
Point Estimate
Upper
Confidence
Limit
Width of
confidence interval
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Point Estimates
We can estimate a
Population Parameter…
Mean
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μ
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with a Sample
Statistic (Point Estimate).
X
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Confidence Interval Estimate
• An interval gives a range of values:
o Takes into consideration variation in sample statistics
from sample to sample.
o Based on observations from 1 sample.
o Gives information about closeness to unknown
population parameters.
o Stated in terms of level of confidence.
o Can never be 100% confident.
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Estimation Process
Random Sample
Population
Mean
X = 50
(mean, μ, is
unknown)
I am 95%
confident that
μ is between
40 & 60.
Sample
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Confidence Interval
• The general formula for all confidence intervals is:
Point Estimate ± (Critical Value)*(Standard Error)
Represents confidence for which the interval will contain
the unknown population parameter.
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Confidence Level (1-)
• Common confidence levels = 90%, 95% or 99%:
o Also written (1 - ) = 0.90, 0.95 or 0.99
• A relative frequency interpretation:
o In the long run, 90%, 95% or 99% of all the confidence
intervals that can be constructed (in repeated samples)
will contain the unknown true parameter.
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Confidence Level (1-) 2
• A specific interval will either contain or will not contain the
true parameter.
o No probability involved in a specific interval.
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Confidence Intervals
Confidence
Intervals
Population
Population
Mean
Proportion
(not part of this
course)
σ Known
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σ Unknown
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Confidence Interval for μ (σ Known)
•
Assumptions:
o Population standard deviation σ is known
o Population is normally distributed
o If population is not normal, check n and hence CLT.
•
Confidence interval estimate
σ
XZ
n
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Confidence Interval for μ (σ Known)2
Where X is the point estimate
Z is the normal distribution critical
σ/ n value for a probability of /2 in each tail
is the standard error
σ
XZ
n
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Finding the Critical Z Value
• Consider a 95% confidence interval
Z  1.96
1   0.95
α
 0.025
2
α
 0.025
2
Z units: Z= -1.96
X units:
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Lower
Confidence
Limit
0
Point Estimate
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Z= 1.96
Upper
Confidence
Limit
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Common Levels of Confidence
90%, 95% and 99%
Confidence
Level
80%
90%
95%
98%
99%
99.8%
99.9%
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Confidence
Coefficient
1 
0.80
0.90
0.95
0.98
0.99
0.998
0.999
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Z value
1.28
1.645
1.96
2.33
2.575
3.08
3.27
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Intervals and Level of Confidence
Sampling Distribution of the Mean
/2
Intervals
extend from
σ
XZ
n
to
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σ
XZ
n
1 
/2
x
μx  μ
X1
X
X
2
3
X
4
Xn
Confidence Intervals
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(1-)x100%
of intervals
constructed
contain μ;
()x100%
do not
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Confidence Interval Example
• A sample of 11 pizza shops from a large normal population
has a mean pizza cooking time of 2.20 minutes. We know
from past testing that the population standard deviation is
0.35 minutes.
• Determine a 95% confidence interval for the true mean pizza
cooking time of the population.
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Confidence
Interval
Example 2
• We are 95% confident that the true
mean cooking time is between
1.9932 and 2.4068 minutes.
• Although the true mean may or may
not be in this interval, 95% of
XZ
σ
n
intervals formed in this manner
(in repeated samples) will contain
 2.20  1.96 (0.35/ 11 )
the true mean.
 2.20  0.2068
 1.9932    2.4068
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Confidence Intervals
Confidence
Intervals
Population
Population
Mean
Proportion
(not part of the
course)
σ Known
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σ Unknown
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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.
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Confidence Interval for μ (σ
Unknown) 2
• So we use the Student t distribution instead of the normal
distribution:
o The t value depends on degrees of freedom denoted
by sample size minus 1 i.e. (d.f = n - 1).
o d.f are number of observations that are free to vary after
sample mean has been calculated.
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Confidence Interval for μ (σ
Unknown) 3
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.
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Student’s t Distribution
• Note: t
Z as n increases
Standard
Normal
(t with d.f = ∞)
t distributions are bellshaped and symmetric,
but have ‘fatter’ tails than
the normal
t (d.f = 25)
t (d.f = 5)
0
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t
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Student’s Table
Upper Tail Area
d.F .25
.10
.05
1
1.000 3.078 6.314
2
3
0.817 1.886 2.920
0.765 1.638 2.353
The body of the table
contains t values, not
probabilities.
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Let: n = 3
d.f = n - 1 = 2
 = 0.10
/2 = 0.05
/2 = 0.05
0
2.920
t
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Confidence Interval Example
• A random sample of 25 businesswomen has a mean age of
50 and standard deviation of 8. Form a 95% confidence
interval for μ.
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Confidence Interval Example 2
• d.F = 25-1 = 24
t/2 , n1  t 0.025,24  2.0639
X  t /2, n-1
S
8
 50  (2.0639)
n
25
46.698 ≤ μ ≤ 53.302
We are 95% confident that the true mean age of
businesswoman is between 46.698 and 53.305 years.
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Determining Sample Size
• If the Sample Size is too big it requires too much resources
($$$)
• If the Sample Size is too small it won’t do the job
What would be the minimum required sample size?
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Sampling Error
• The required sample size can be found to reach a desired
margin of error (e) with a specified level of confidence (1 ).
• The margin of error is also called a sampling error.
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Sampling Error 2
o The amount of imprecision in the estimate of the
population parameter.
o The amount added and subtracted to the point estimate to
form the confidence interval.
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Determining Sample Size for the
Mean
• Confidence interval formula for population mean
σ
XZ
n
Sampling error
(margin of error)
σ
eZ
n
denoted e
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Determining Sample Size for the
Mean 2
σ
eZ
n
Z σ
n
2
e
2
Now solve
for n to get
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Determining Sample Size for the
Mean 3
• To determine the required sample size for the mean,
you must know:
o The desired level of confidence (1 - ), which
determines the critical Z value.
o The acceptable sampling error, e.
o The standard deviation, σ.
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Determining Sample Size for the
Mean 4
If  = 45, what sample size is needed to estimate the mean
within ± 5 with 90% confidence?
Z σ
(1.645) (45)
n

 219.19
2
2
e
5
2
2
2
2
So the required sample size is n = 220
(Always round up)
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If σ is Unknown
• If unknown, σ can be estimated:
o From past data using that data’s standard deviation.
o If population is normal, range is approximately 6σ so we
can estimate σ by dividing the range by 6.
o Conduct a pilot study and estimate σ with the sample
standard deviation, s.
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Ethical Issues
• A confidence interval estimate (reflecting sampling error)
should always be included when reporting a point estimate.
• The level of confidence should always be reported.
• The sample size should be reported.
• An interpretation of the confidence interval estimate should
also be provided.
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Topic Summary
• Discussed point estimates.
• Developed confidence interval estimates:
o for the mean (σ known)
o for the mean (σ unknown)
• Determined required sample size for mean
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