Confidence Intervals
Estimation
 Department of ISM, University of Alabama, 1995-2003
M33 Confidence intervals
1
Lesson Objective
 Learn how to construct a
confidence interval estimate
for many situations.
 L.O.P.
 Understand the meaning
of being “95%” confident
by using a simulation.
 Learn how confidence intervals
are used in making decisions
about population parameters.
 Department of ISM, University of Alabama, 1995-2003
M33 Confidence intervals
2
Statistical Inference
Generalizing from a sample
to a population,
by using a statistic
to estimate
a parameter.
Goal: To make a decision.
 Department of ISM, University of Alabama, 1995-2003
M33 Confidence intervals
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Statistical Inference
 Estimation of parameter:
1. Point estimators
2. Confidence intervals
 Testing parameter values using:
1. Confidence intervals
2. p-values
3. Critical regions.
 Department of ISM, University of Alabama, 1995-2003
M33 Confidence intervals
4
Confidence Interval
point estimate ± margin of error
Choose the appropriate statistic
and its corresponding m.o.e.
based on the problem that is to
be solved.
 Department of ISM, University of Alabama, 1995-2003
M33 Confidence intervals
5
Estimation of Parameters
A (1-a)100% confidence interval estimate of a parameter is
point estimate  m.o.e.
Population
Parameter
Point Estimator
Mean, m
if s is known:
Mean, m
if s is unknown:
m.o.e. = Zα
x
m.o.e. = t( α
x
^
pp  X / n,
Proportion, p:
Margin of Error
at (1-a)100% confidence
m.o.e. = Zα
x1  x2
m.o.e. = Z α
Diff. of two
proportions, p1 - p2 :
pˆ1  pˆ 2
m.o.e. = Zα
Mean from a
regression
when X = x*:
, n-1)
2
2
s
n
n
ˆ ˆ n
p(1-p)
s12 s22
+
n1 n2
pˆ1 (1-pˆ1 ) pˆ2 (1-pˆ2 )
+
2
n1
n2
m.o.e. = t( α , n-2) s
2
Equ.2
b
where s 
yˆ  a  bx *
2
2
Diff. of two
means, m1 - m2 :
(for large sample sizes only)
Slope of regression
line, b :
σ
MSE
m.o.e. =t( α
2
, n-2)
1 (x * -x)2
s
+
n Equ.2
Estimation of Parameters
A (1-a)100% confidence interval estimate of a parameter is
point estimate  m.o.e.
Population
Parameter
Point Estimator
Mean, m
if s is known:
x
Mean, m
if s is unknown:
x
Proportion, p:
Margin of Error
at (1-a)100% confidence
m.o.e.  Za  s
2
^
pp  X / n,
m.o.e.  t( a
n
s

, n 1)
2
n
m.o.e.  Za  pˆ (1 pˆ ) n
2
When is the population of
all possible X values Normal?
 Anytime the original pop.
is Normal,
(“exactly” for any n).
 Anytime the original pop.
is not Normal, but
n is BIG; (n > 30).
 Department of ISM, University of Alabama, 1995-2003
M33 Confidence intervals
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Margin of Error for 95% confidence:
=
1.96 l
s
n
To get a smaller Margin of Error:


 Department of ISM, University of Alabama, 1995-2003
M33 Confidence intervals
9
Confidence Intervals
point estimate ± margin of error
Estimate the true mean net weight of
16 oz. bags of Golden Flake Potato Chips
with a 95% confidence interval.
Data:
s = .24 oz. (True population standard deviation.)
Sample size = 9.
Must assume
Sample mean = 15.90 oz.
ori. pop. is
Normal .
Distribution of individual bags is ______
 Department of ISM, University of Alabama, 1995-2003
M33 Confidence intervals 10
For 95% confidence
when s is known:
m.o.e. =
s = .24 oz.
n = 9.
X = 15.90 oz.
95% confidence interval for m:
15.90 
 Department of ISM, University of Alabama, 1995-2003
M33 Confidence intervals 11
Statement in the L.O.P.
“I am 95% confident that
the true mean net weight of
Golden Flake 16 oz. bags of potato chips
falls in the interval 15.5472 to 16.2528 oz.”
A statement in L.O.P. must contain four parts:
1. amount of confidence.
2. the parameter being estimated in L.O.P.
3. the population to which we generalize
in L.O.P.
4. the calculated interval.
 Department of ISM, University of Alabama, 1995-2003
M33 Confidence intervals 12
Meaning of being 95% Confident
If we took many, many, samples
from the same population,
under the same conditions, and we
constructed a 95% CI from each,
then we would expect that
95% of all these many, many
different confidence intervals
would contain the true mean,
and 5% would not.
 Department of ISM, University of Alabama, 1995-2003
M33 Confidence intervals 13
Reality: We will take only ONE sample.
l
15.7
15.9
16.1
m.o.e.
X
+ m.o.e.
X-axis
Is the true population mean in this interval?
I cannot tell with certainty;
but I am 95% confident it does.
Making a decision using a CI.
Question of interest: Is there evidence
that the hypothesized mean is not true,
at the “a” level of significance?
 If the “hypothesized value” is inside the CI,
this value may be a plausible value.
Make a vague conclusion.
 If the “hypothesized value” is not in the CI,
this value IS NOT a plausible value.
Reject it! Make a strong conclusion.
Take appropriate action!
Confidence level
= 1
Level of significance
=
 Department of ISM, University of Alabama, 1995-2003
a
a
= .95
= .05
M33 Confidence intervals 16
p2
The “true” population
mean is hypothesized
to be 13.0.
0.5
Population of
all possible
X-bar values,
assuming . . . .
0.4
Conclusion:
The hypothesis is
wrong. The “true”
mean not 13.0!
0.3
0.2
0.1
My ONE
sample mean.
0.0
Middle
95%
l
l
5
8
l
11
5.6 7.9 10.2
My ONE
Confidence Interval.
14
X
17
20
X-axis
23
13.0 does NOT fall in
my confidence interval;
 it is not a plausible mean.
 Department of ISM, University of Alabama, 1995-2003
M33 Confidence intervals 17
p1
0.5 data
The
convince
0.4 the
me
A more likely location
of the population.
The “true” population
mean is hypothesized
to be 13.0.
true
0.3
mean
is
Conclusion:
The hypothesis is
wrong. The “true”
mean not 13.0!
smaller
0.2 13.0,
than
around
0.1
7.9!
0.0
l
5
5.6
8
l
l
11
7.9 10.2
14
X
17
20
X-axis
23
13.0 does NOT fall in
my confidence interval;
 it is not a plausible mean.
 Department of ISM, University of Alabama, 1995-2003
M33 Confidence intervals 18
Net weight of potato chip bags
should be 16.00 oz.
FDA inspector takes a sample.
If 95% CI is, say, (15.81 to 15.95), X = 15.88
then 16.00 is NOT in the interval.
Therefore, reject 16.00 as a plausible
value. Take action against the company.
If 95% CI is, say, (15.71 to 16.05), X = 15.88
then 16.00 IS in the interval.
Therefore, __________________________
___________________________________
Net weight of potato chip bags
should be 16.00 oz.
FDA inspector takes a sample.
If 95% CI is, say, (16.05 to 16.15),
then 16.00 is NOT in the interval.
X = 16.10
Therefore, __________________________.
But, the FDA does not care that
the company is giving away potato chips.
The FDA would obviously _____________
against the company
Meaning of being 95% Confident
If we took many, many, samples
from the same population,
under the same conditions, and we
constructed a 95% CI from each,
then we would expect that
95% of all these many, many
different confidence intervals
would contain the true mean,
and 5% would not.
 Department of ISM, University of Alabama, 1995-2003
M33 Confidence intervals 21
Interpretation of “Margin of Error”
A sample mean X calculated from a
simple random sample has
a 95% chance of being “within the
range of the true population mean, m,
plus and minus the margin of error.”
True - m.o.e.
mean
True
mean
True
mean + m.o.e.
A sample mean is likely to fall in this
interval, but it may not.
 Department of ISM, University of Alabama, 1995-2002
M32 Margin of Error
22
A common misconception
m  m.o.e.
X  m.o.e.
This region
DOES NOT.
This region DOES contain
95% of all possible X-bars.
.95
-4.0
-3.0
-2.0
-1.0
m
0.0
1.0
2.0
-4.0
3.0
X-axis
-3.0
4.0
-2.0
-1.0
Xm
0.0
1.0
2.0
3.0
4.0
X-axis
A random x-bar
 Department of ISM, University of Alabama, 1995-2002
M32 Margin of Error
23
Concept questions.
Our 95% confidence interval is
15.7 to 16.1.
X = 15.9
Is our confidence interval one of
the 95%, or one of the 5%?
Yes or No or ?
Does the true population mean
lie between 15.7 and 16.1?
Does the sample mean
lie between 15.7 and 16.1?
What is the probability that
m lies between 15.7 and 16.1?
 Department of ISM, University of Alabama, 1995-2003
M33 Confidence intervals 24
Concept questions.
Our 95% confidence interval is
15.7 to 16.1.
X = 15.9
Yes or No or ?
Do 95% of the sample data lie
between 15.7 and 16.1?
Is the probability .95 that a future sample
mean will lie between 15.7 and 16.1?
Do 95% of all possible sample means
lie between m e.m. and m + e.m.?
If the confidence level is higher,
will the interval width be wider?
 Department of ISM, University of Alabama, 1995-2003
M33 Confidence intervals 25
End of File M33
 Department of ISM, University of Alabama, 1995-2003
M33 Confidence intervals 26