Confidence Interval Estimation

M33 Confidence intervals 1 Department of ISM, University of Alabama, 1995-2003

Confidence IntervalConfidence Interval

EstimationEstimation


Lesson ObjectiveLesson Objective Learn how to Learn how to constructconstruct a a

confidence interval estimate confidence interval estimate for many situations.for many situations.

L.O.P.L.O.P. Understand the Understand the meaningmeaning

of being “95%” confident of being “95%” confident by using a simulation.by using a simulation.

Learn how confidence intervalsLearn how confidence intervalsare used in are used in making decisionsmaking decisionsabout population parameters.about population parameters.


Statistical Inference

Generalizing from a sample to a population,

by using a statisticto estimate

a parameter.

Goal: Goal: To make a decision.To make a decision.


Estimation of parameter: 1. Point estimators 2. Confidence intervals

Statistical Inference

Testing parameter values using: 1. Confidence intervals

2. p-values 3. Critical regions.


Confidence Interval

point estimate ± margin of error point estimate ± margin of error

Choose the appropriate Choose the appropriate statisticstatisticand its corresponding and its corresponding m.o.e.m.o.e.based on the problem that is tobased on the problem that is tobe solved.be solved.

Diff. of two proportions, p1 - p2 :

A (1-A (1-)100% confidence interval estimate of a parameter is)100% confidence interval estimate of a parameter is

21 ˆˆ pp 1 1 2 2α2 1 2

p (1-p p (1-pm.o.e. = Z +

n n

ˆ ˆ ˆ ˆ) )

21 xx 2 21 2

α2 1 2

s sm.o.e. = Z +

n n

Proportion, p: α2

m.o.e. = Z p(1-p) nˆ ˆ

α2

σm.o.e. = Zn

Mean, if is known:

α( , n-1)2

sm.o.e. = tn

PopulationParameter Point Estimator Margin of Error

at (1-)100% confidence

Mean, if is unknown:

/ ,p X n

x

x

point estimatepoint estimate m.o.e.m.o.e.

Estimation of Parameters

Diff. of two means, 1 - 2 :

(for large sample sizes only)

α( , n-2)2

sm.o.e. = tEqu.2

Slope of regression line, :

b

* 2

α( , n-2)2

1 (x -x)m.o.e.=t s +

n Equ.2

Mean from a regression when X = x*:

*ˆ bxay where s MSE

p̂

Proportion, :

Mean, if is known:

PopulationParameter Point Estimator Margin of Error

at (1-)100% confidence

Mean, if is unknown:

2

ˆ ˆ. . . (1 )mo e Z p p n

2. . .mo e Z

n

( , 1)2. . .

nsmo e tn

/ ,p X n

x

x

p̂

A (1-A (1-)100% confidence interval estimate of a parameter is)100% confidence interval estimate of a parameter is

point estimatepoint estimate m.o.e.m.o.e.

Estimation of Parameters

The theory that supports this


requires that the population


of all possible X’s is


normally distributed.











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).


Confidence Intervalspoint estimate ± margin of error point estimate ± margin of error

Estimate the true mean net weight of Estimate the true mean net weight of 16 oz. bags of Golden Flake Potato Chips 16 oz. bags of Golden Flake Potato Chips with a 95% confidence interval. with a 95% confidence interval. Data:Data:

= .24 oz. = .24 oz. (True population standard deviation.)(True population standard deviation.)

Sample size = 9.Sample size = 9.

Sample mean = 15.90 oz.Sample mean = 15.90 oz.

Distribution of individual bags is ______ .Distribution of individual bags is ______ .Must assumeMust assumeori. pop. is ori. pop. is

NormalNormal

Must assumeMust assumeori. pop. is ori. pop. is

NormalNormal


For 95% confidencewhen is known:

= = 1.96 .24 / 3

= .3528 oz..3528 oz.

= .24 oz. = .24 oz.

n = 9.n = 9.

X = 15.90 oz.X = 15.90 oz.

m.o.e. = 1.96 n

95% confidence interval for :15.90 .3528

15.5472 to 16.2528 ounces15.5472 to 16.2528 ounces

ZZ.025.025 == 1.961.96


““I am 95% confident thatI am 95% confident thatthe true mean net weight ofthe true mean net weight ofGolden Flake 16 oz. bags of potato chips Golden Flake 16 oz. bags of potato chips falls in the interval falls in the interval 15.547215.5472 to to 16.2528 oz16.2528 oz.”.”

Statement in the L.O.P. Statement in the L.O.P.

A statement in A statement in L.O.P.L.O.P. must contain four parts: must contain four parts: 1. amount of 1. amount of confidenceconfidence.. 2. the 2. the parameterparameter being estimated in being estimated in L.O.P.L.O.P. 3. the 3. the populationpopulation to which we generalize to which we generalize in in L.O.P.L.O.P. 4. the calculated 4. the calculated intervalinterval. .


Simulation to Illustrate Simulation to Illustrate the meaningthe meaning

of a confidence intervalof a confidence interval

Simulation to Illustrate Simulation to Illustrate the meaningthe meaning

of a confidence intervalof a confidence interval


Find the interval around the mean in which 95% of all possible sample means fall.

.9500 .0250.0250

Xm.o.e.m.o.e.- - m.o.e.m.o.e. -axis-axis



.9500 .0250.0250

X

-axis-axism.o.e.m.o.e.- - m.o.e.m.o.e.



.9500 .0250.0250

Xm.o.e.- m.o.e.

-axis-axis


.9500 .0250.0250

Xm.o.e.- m.o.e.

95% of the intervals willcontain , 5% will not.

95% of the intervals willcontain , 5% will not.

-axis-axis


.9500 .0250.0250

Xm.o.e.- m.o.e.

SimulationSimulationSimulationSimulation

-axis-axis


114 of 120 CI’s (95%) contain ,

6 of 120 CI’s ( 5%) do not.

114 of 120 CI’s (95%) contain ,

6 of 120 CI’s ( 5%) do not.

.9500 .0250.0250

Xm.o.e.- m.o.e.

SimulationSimulationSimulationSimulation

-axis-axis


Meaning of being 95% Confident

If we took If we took many, many,many, many, samples samples from the same population, from the same population, under the same conditions, and weunder the same conditions, and weconstructed a 95% CI from each,constructed a 95% CI from each,

then we would then we would expectexpect that that 95%95% of all these of all these many, manymany, many different confidence intervals different confidence intervals would containwould contain the true mean, the true mean,and and 5% would not5% would not..

Reality: We will take only ONE sample.

X-axis

+ m.o.e.+ m.o.e. m.o.e. m.o.e.

Is the true population mean in this interval?Is the true population mean in this interval?

I cannot tell with certainty; I cannot tell with certainty;

but I am 95% confident it does. but I am 95% confident it does.

XX

65.7 66.165.9

Hypothesized meanHypothesized mean

Making a decision using a CI.Making a decision using a CI.Making a decision using a CI.Making a decision using a CI.

A value of the parameter thatA value of the parameter that we believe is, we believe is, or ought to be or ought to bethe the true valuetrue value of the mean. of the mean.

We gather evidence and make a We gather evidence and make a decision about this hypothesis.decision about this hypothesis.

Question of interest:Question of interest: Is there evidence Is there evidence that the that the true meantrue mean is different than the is different than the hypothesized meanhypothesized mean??

Making a decision using a CI.Making a decision using a CI.Making a decision using a CI.Making a decision using a CI.

If the “hypothesized value” If the “hypothesized value” is insideis inside the CI, the CI,

then this then this ISIS a plausible value. a plausible value. Make a Make a vaguevague conclusion. conclusion.

If the “hypothesized value” If the “hypothesized value” is notis not in the CI, in the CI,

then this then this IS NOTIS NOT a plausible value. a plausible value.Reject it! Make a Reject it! Make a strongstrong conclusion. conclusion.Take appropriate action!Take appropriate action!


Confidence level = 1 Confidence level = 1

Level of significance = Level of significance = = .95= .95

= .05= .05


The “true” populationThe “true” population mean is hypothesized mean is hypothesized

to be 13.0.to be 13.0.

X-axis

Population ofPopulation ofall possibleall possible

X-bar values,X-bar values,assuming . . . .assuming . . . .

My ONEMy ONEsample mean.sample mean.

My My ONEONEConfidence Interval.Confidence Interval.

Conclusion:The hypothesis is wrong. The “true”

mean not 13.0!

13.0 does NOT fall in 13.0 does NOT fall in my confidence interval; my confidence interval;

it is it is notnot a plausible value a plausible valuefor the true mean. for the true mean.

10.210.25.65.6 7.97.9

MiddleMiddle95%95%

The data convince me the

true mean is smaller

than 13.0.

I am 95% confidentthat . . . .


The “true” populationThe “true” population mean is hypothesized mean is hypothesized

to be 13.0.to be 13.0.

Conclusion:The hypothesis is wrong. The “true”

mean not 13.0!

10.210.25.65.6 7.97.9

X-axis

A more likely locationof the population.

13.0 does NOT fall in 13.0 does NOT fall in my confidence interval; my confidence interval;

it is it is notnot a plausible value a plausible valuefor the true mean. for the true mean.

The data convince me the

true mean is smaller

than 13.0.

I am 95% confidentthat . . . .

Net weight of potato chip bagsshould be 16.00 oz.FDA inspector takes a sample.

If 95% CI is, say, If 95% CI is, say, (15.81 to 15.95),(15.81 to 15.95),

If 95% CI is, say, If 95% CI is, say, (15.71 to 16.05),(15.71 to 16.05),

then 16.00 then 16.00 is NOTis NOT in the interval. in the interval.

Therefore, Therefore, rejectreject 16.00 as a plausible 16.00 as a plausiblevalue. value. Take action against the company.Take action against the company.

X = 15.88

then 16.00 then 16.00 ISIS in the interval.in the interval.

Therefore, 16.00 may be a plausibleTherefore, 16.00 may be a plausiblevalue. value. Take no action.Take no action.

X = 15.88

Net weight of potato chip bagsshould be 16.00 oz.FDA inspector takes a sample.

then 16.00 then 16.00 is NOTis NOT in the interval. in the interval.

Therefore, Therefore, rejectreject 16.00 as a plausible 16.00 as a plausiblevalue. value. But, the FDA does not care thatBut, the FDA does not care thatthe company is giving away potato chips.the company is giving away potato chips.

The FDA would obviously take no action The FDA would obviously take no action against the company.against the company.

X = 16.10If 95% CI is, say, If 95% CI is, say, (16.05 to 16.15),(16.05 to 16.15),


Meaning of being 95% Confident

If we took If we took many, many,many, many, samples samples from the same population, from the same population, under the same conditions, and weunder the same conditions, and weconstructed a 95% CI from each,constructed a 95% CI from each,then we would expect that then we would expect that 95%95% of all these of all these many, manymany, many different confidence intervals different confidence intervals would containwould contain the true mean, the true mean,and and 5% would not5% would not..

Recall

M32 Margin of Error 32 Department of ISM, University of Alabama, 1995-2002

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, plus and minus the margin of error.”

Truemean

Truemean+ m.o.e.True

mean - m.o.e.

A sample mean is likely to fall in thisinterval, but it may not.


Is our confidence interval one ofIs our confidence interval one ofthe 95%, or one of the 5%?the 95%, or one of the 5%?

I cannot tellI cannot tellwith certainty!with certainty!

Does the true population mean Does the true population mean lie between 15.7 and 16.1?lie between 15.7 and 16.1?

I cannot tellI cannot tellwith certainty!with certainty!

Does the sample mean Does the sample mean lie between 15.7 and 16.1?lie between 15.7 and 16.1? Yes, dead center!Yes, dead center!

What is the probability that What is the probability that lies between 15.7 and 16.1? lies between 15.7 and 16.1? Zero or One!Zero or One!

Concept questions.Concept questions.Our 95% confidence interval is 15.7 to 16.1. X = 15.9

Yes or No or ?Yes or No or ?


Does 95% of the sample data lie Does 95% of the sample data lie between 15.7 and 16.1?between 15.7 and 16.1?

NO!NO!

If the confidence level is higher,If the confidence level is higher,will the interval width be wider?will the interval width be wider? Yes!Yes!

Is the probability .95 that a Is the probability .95 that a futurefuture sample sample mean will lie between 15.7 and 16.1?mean will lie between 15.7 and 16.1?

NO!NO!

Do 95% of all possible sample means Do 95% of all possible sample means lie between lie between m.o.e. m.o.e. and and + m.o.e. + m.o.e.?? Yes!Yes!

Concept questions.Concept questions.Our 95% confidence interval is 15.7 to 16.1. X = 15.9

Yes or No or ?Yes or No or ?

M31- Dist of X-bars 35 Department of ISM, University of Alabama, 1992-2003

1009080706050403020100X

Original PopulationOriginal Population: Normal (: Normal ( = 50, = 50, = 18) = 18)

n = 36 = 3.00x

= 18.00

Confidence Interval Estimation

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