# M33- Confidence Intervals Handout by hedongchenchen

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```									  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   3
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                Point Estimator                 Margin of Error
Parameter                                             at (1-a)100% confidence
Mean, m                              x                 m.o.e. = Z α                     σ
if s is known:                                                               2             n
Mean, m                              x                 m.o.e. = t( α                    s
if s is unknown:                                                         2
, n-1)
n
Proportion, p:                  ^  X / n,

p
p                      m.o.e. = Z α                    ˆ ˆ
p(1-p) n
2

Diff. of two                                                                                2
s12 s2
means, m1 - m2 :                 x1  x2               m.o.e. = Zα                        +
(for large sample sizes only)
2            n1 n2
Diff. of two                                                               ˆ      ˆ    ˆ     ˆ
p1 (1-p1 ) p2 (1-p2 )
p1  p2
ˆ    ˆ                m.o.e. = Zα                   +
proportions, p1 - p2 :                                              2          n1         n2
Slope of regression              b                     m.o.e. = t( α , n-2) s
line, b :                                                          2              Equ.2
where s     MSE
Mean from a
y  a  bx *
ˆ                                                             1 (x * -x)2
regression                                            m.o.e. =t( α                    s  +
when X = x*:                                                         2
, n-2)
n Equ.2
Estimation of Parameters
A (1-a)100% confidence interval estimate of a parameter is
point estimate  m.o.e.

Population                                     Margin of Error
Parameter         Point Estimator
at (1-a)100% confidence

Mean, m
if s is known:
x                 m.o.e.  Za  s
2       n

Mean, m                    x                m.o.e.  t( a    , n 1)
s
if s is unknown:                                            2            n

Proportion, p:        ^  X / n,

p
p                  m.o.e.  Za  p(1  p ) n
ˆ     ˆ
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   8
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.
Sample mean = 15.90 oz.                Must assume
ori. pop. is
Normal
Distribution of individual bags is ______ .
 Department of ISM, University of Alabama, 1995-2003   M33 Confidence intervals 10
s = .24 oz.
For 95% confidence
when s is known:                                        n = 9.
X = 15.90 oz.
m.o.e. =

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              X-axis
- m.o.e.      X       + m.o.e.

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-a           = .95

Level of significance                                =    a          = .05

 Department of ISM, University of Alabama, 1995-2003    M33 Confidence intervals 16
0.5                                               The “true” population
Population of                     mean is hypothesized
all possible                          to be 13.0.
p2

0.4
X-bar values,
assuming . . . .
0.3                                                         Conclusion:
The hypothesis is
0.2                                                      wrong. The “true”
My ONE                                          mean not 13.0!
sample mean.
0.1
Middle
95%
0.0                 l                 l

5         8         11          14       17       20    X-axis
23
l
5.6 7.9 10.2                  X            13.0 does NOT fall in
My ONE                                  my confidence interval;
Confidence Interval.                       it is not a plausible mean.

 Department of ISM, University of Alabama, 1995-2003        M33 Confidence intervals 17
A more likely location
0.5
The data         of the population.                The “true” population
convince                                            mean is hypothesized
to be 13.0.
p1

me the
0.4
true
0.3                                                           Conclusion:
mean is                                                    The hypothesis is
smaller                                                   wrong. The “true”
0.2
than 13.0,                                                  mean not 13.0!
around
0.1
7.9!
0.0                 l                 l

5         8         11          14       17       20    X-axis
23
l
5.6    7.9 10.2            X            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), X = 16.10
then 16.00 is NOT in the interval.
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.                    True            True
mean                             mean            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 contain                                             This region
95% of all possible X-bars.                                          DOES NOT.

.95

m                                               Xm
-4.0    -3.0   -2.0   -1.0    0.0   1.0   2.0    -4.0
3.0      -3.0
4.0     -2.0   -1.0     0.0   1.0   2.0   3.0   4.0

X-axis                                                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
Yes or No or ?
Is our confidence interval one of
the 95%, or one of the 5%?
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

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