Confidence Interval Estimation

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Confidence IntervalEstimation
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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 intervalsare used in making
  decisionsabout population parameters.

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Statistical Inference
Generalizing from a sample to a population,by
using a statisticto estimatea parameter.
Goal To make a decision.
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Statistical Inference

• q Estimation of parameter 1. Point
  estimators 2. Confidence intervals

q Testing parameter values using 1.
Confidence intervals 2. p-values
3. Critical regions.
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Confidence Interval
point estimate margin of error
Choose the appropriate statisticand its
corresponding m.o.e.based on the problem that is
tobe solved.
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Estimation of Parameters
A (1-a)100 confidence interval estimate of a
parameter is
point estimate m.o.e.
Margin of Error at (1-a)100 confidence
PopulationParameter
Point Estimator
Mean, m if s is known
Mean, m if s is unknown
Proportion, p
Diff. of two means, m1 - m2 (for large sample
sizes only)
Diff. of two proportions, p1 - p2
Slope of regression line, b
Mean from a regression when X x
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Estimation of Parameters
A (1-a)100 confidence interval estimate of a
parameter is
point estimate m.o.e.
Margin of Error at (1-a)100 confidence
PopulationParameter
Point Estimator
Mean, m if s is known
Mean, m if s is unknown

Proportion, p
p
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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 gt 30).

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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.
Distribution of individual bags is ______ .
Must assumeori. pop. is Normal
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s .24 oz. n 9. X 15.90 oz.
For 95 confidencewhen s is known
1.96
1.96 l .24 / 3 .3528 oz.
95 confidence interval for m
15.90 ? .3528
15.5472 to 16.2528 ounces
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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.
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Simulation to Illustrate the meaningof a
confidence interval
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Find the interval around the mean in which 95 of
all possible sample means fall.
m m.o.e.
m - m.o.e.
m
-axis
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Find the interval around the mean in which 95 of
all possible sample means fall.
l
m
m m.o.e.
m - m.o.e.
-axis
l
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Find the interval around the mean in which 95 of
all possible sample means fall.
l
-axis
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Find the interval around the mean in which 95 of
all possible sample means fall.
l
-axis
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Find the interval around the mean in which 95 of
all possible sample means fall.
l
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Find the interval around the mean in which 95 of
all possible sample means fall.
l
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Find the interval around the mean in which 95 of
all possible sample means fall.
.0250
.9500
.0250
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m
m m.o.e.
m - m.o.e.
-axis
l
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95 of the intervals willcontain m , 5 will
not.
l
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Find the interval around the mean in which 95 of
all possible sample means fall.
Simulation
.0250
.9500
.0250
m
m m.o.e.
m - m.o.e.
-axis
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Find the interval around the mean in which 95 of
all possible sample means fall.
l
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Simulation
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m m.o.e.
m - m.o.e.
-axis
114 of 120 CIs (95) contain m , 6 of 120
CIs ( 5) do not.
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Meaning of being 95 Confident
If we took many, many, samples from the same
population, under the same conditions, and
weconstructed 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.
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Reality We will take only ONE sample.
l
65.7
66.1
65.9
m.o.e.
m.o.e.
Is the true population mean in this interval?
I cannot tell with certainty but I
am 95 confident it does.
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Making a decision using a CI.
Hypothesized mean
A value of the parameter that we believe
is, or ought to bethe true value of the
mean.
We gather evidence and make a decision about
this hypothesis.
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Making a decision using a CI.
Question of interest Is there evidence that
the true mean is different than the hypothesized
mean?
q If the hypothesized value is inside the CI,
then this IS a plausible value. ? Make a vague
conclusion.
q If the hypothesized value is not in the CI,
then this IS NOT a plausible value.Reject it!
Make a strong conclusion.Take appropriate action!
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.95 .05
Confidence level 1 a
Level of significance a
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The data convince me the true mean is
smallerthan 13.0. I am 95 confidentthat . .
. .
The true population mean is hypothesized to
be 13.0.
Population ofall possibleX-bar values,assuming
. . . .
ConclusionThe hypothesis is wrong. The true
mean not 13.0!
My ONEsample mean.
Middle95
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13.0 does NOT fall in my confidence interval?
it is not a plausible valuefor the true mean.
10.2
7.9
5.6
My ONEConfidence Interval.
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A more likely locationof the population.
The data convince me the true mean is
smallerthan 13.0. I am 95 confidentthat . .
. .
The true population mean is hypothesized to
be 13.0.
ConclusionThe hypothesis is wrong. The true
mean not 13.0!
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13.0 does NOT fall in my confidence interval?
it is not a plausible valuefor the true mean.
10.2
7.9
5.6
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Net weight of potato chip bagsshould be 16.00
oz.FDA inspector takes a sample.
If 95 CI is, say, (15.81 to 15.95),
then 16.00 is NOT in the interval. Therefore,
reject 16.00 as a plausiblevalue. Take action
against the company.
If 95 CI is, say, (15.71 to 16.05),
then 16.00 IS in the interval. Therefore,
16.00 may be a plausiblevalue. Take no action.
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Net weight of potato chip bagsshould 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. Therefore,
reject 16.00 as a plausiblevalue. But, the FDA
does not care thatthe company is giving away
potato chips. The FDA would obviously take no
action against the company.
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Meaning of being 95 Confident
If we took many, many, samples from the same
population, under the same conditions, and
weconstructed 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.
Recall
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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.

Truemean
A sample mean is likely to fall in thisinterval,
but it may not.
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Concept questions.Our 95 confidence interval
is 15.7 to 16.1.
Yes or No or ?
Is our confidence interval one of the 95, or one
of the 5?
I cannot tellwith certainty!
Does the true population mean lie between 15.7
and 16.1?
I cannot tellwith certainty!
Does the sample mean lie between 15.7 and 16.1?
Yes, dead center!
What is the probability that m lies between 15.7
and 16.1?
Zero or One!
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Concept questions.Our 95 confidence interval
is 15.7 to 16.1.
Yes or No or ?
Does 95 of the sample data lie between 15.7 and
16.1?
NO!
Is the probability .95 that a future sample mean
will lie between 15.7 and 16.1?
NO!
Do 95 of all possible sample means lie between
m m.o.e. and m m.o.e.?
Yes!
If the confidence level is higher,will the
interval width be wider?
Yes!
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Original Population Normal (m 50, s 18)
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s 18.00