Chapter 3 - Sampling Distribution and Confidence Interval1

8/8/2019 Chapter 3 - Sampling Distribution and Confidence Interval1

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SAMPLINGSAMPLING

DISTRIBUTIONSDISTRIBUTIONS& CONFIDENCE& CONFIDENCE

INTERVALINTERVALCHAPTER 3BUM 2413 / BPF 3313


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CONTENTCONTENT

3.1 Sampling Distribution

3.2 Estimate, Estimation and Estimator

3.3 Confidence Interval for the mean

3.4 Confidence Interval for the Difference

between Two mean3.5 Confidence Interval for the Proportion

3.6 Confidence Interval for the Difference

between Two Proportions

3.7 Confidence Interval for Variances and

Standard Deviations

3.8 Confidence Interval for Two Variances

and Standard Deviations


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SAMPLING DISTRIBUTION

Asampling distribution is the probability distribution,under repeated sampling of the population, of a givenstatistic (a numerical quantity calculated from the datavalues in a sample

).

The formula for the sampling distribution depends on thedistribution of the population, the statistic beingconsidered, and the sample size used. Amore preciseformulation would speak of the distribution of the statisticfor all possible samples of a given size, not just "underrepeated sampling".

In other word, the sampling distribution of a statistic S forsamples of size n is defined as follows:y The experiment consists of choosing a sample of size n from

the population and measuring the statistic S. The samplingdistribution is the resulting probability distribution.


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EXAMPLE

Imagine that our population consists of only three numbers:

the number 2, the number 3 and the number 4. Our plan is

to draw an infinite number of random samples of size n = 2

and form a sampling distribution of the sample means.


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SAMPLING DISTRIBUTION FOR

MEAN

The mean of the sampling distribution of means is equal tothe population mean.

The standard deviation of the sampling distribution of meansis

for infinite population

for finite population

If the population is normally distributed, the samplingdistribution is normal regardless of sample size.

By using the Central Limit Theorem,

If the population distribution is not necessarily normal, andhas mean and standard deviation , then, for sufficientlylarge n, the sampling distribution of is approximatelynormal, with mean and standard deviation

XQ Q!

X

n

WW !

2

~ ,X Nn

WQ

XQ Q!

X

n

WW !

1XN n

Nn

WW

!

X


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DIFFERENT MEAN

2 2

1 21 1 2 2

1 2

2 2

1 21 2 1 2

1 2

2 21 2

1 2 1 2

1 2

For , and ,

,

,

X N X N n n

X X N n n

X X N n n

W WQ Q

W WQ Q

W WQ Q


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PROPORTIONS

p - Proportion, Probability and Percent forpopulation

- sample proportion ofx successes in a sample of

size n

- sample proportion of failures in a sample of sizen

x is the binomial random variable created by counting the

number of successes picked by drawingn

times from thepopulation.

The shape of the binomial distribution looks fairly Normalas long as n is large and/orp is not too extreme (not close to0 or 1).

x

pn

!

1q p!

~ , X bin np

1If 1 5, then ,

p pnp p p N p

n

"


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PROPORTIONS

The sampling distribution for proportions is a distribution of theproportions of all possible n samples that could be taken in a

given situation.

That is, the sample proportion (percent of successes in a sample),

is approximately Normally distributed with

y meanp, and

y

standard deviation 1p pn


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DIFFERENT PROPORTIONS

1 1 2 2

1 1 2 2

12

1 1 2 2

1 2 1 2

1 2

1 1 2 2

1 2 1 2

1 2

1 1 I ~ , and ~ ,

1 1 ~ ,

1 1 ~ ,

p p p pp N p p N p

n n

p p p pp p N p p

n n

p p p pp p N p p

n n


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ESTIMATOR

Probability function are actually families ofmodels in the sense that each include one ormore parameter.

Example: Poisson, Binomial, Normal

Any function of a random sample whoseobjective is to approximate a parameter is called

a statistic or an estimator

is the estimator forU U

statistic parameter


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PROPERTIES OF GOOD

ESTIMATOR

Unbiased

y

Efficient

y

Sufficient

y

Consistent

y


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ESTIMATIONS & ESTIMATE

Estimation Is the entire process of using anestimator to produce an estimate of theparameter

2 types of estimation

1. Point Estimate Asingle number used to estimate a population

parameter

2. Interval Estimate A

spread of values used to estimate a populationparameter

The interval is usually written (a, b) where a and b areknown as confidence limit

a lower confidence limit

b upper confidence limit


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DEFINITIONS

Confidence Interval

y Range of numbers that have a high probabilityof containing the unknown parameter as an

interior point.y By looking at the width of a confidence

interval, we can get a good sense of theestimator precision.

y Width = b a

Confidence Coefficient ( )

y The probability of correctly including thepopulation parameter being estimated in theinterval that is produced

1 E


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DEFINITIONS

Level of Confidence

y The confidence coefficient expressed as a

percent ,

y Example: 1 100%E v

1 %E


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3.33.3 CONFIDENCE INTERVAL FOR MEAN

OBJECTIVESOBJECTIVES

After completing this chapter, you should be able to

1. Find the confidence interval for the mean.

2. Find the confidence interval for the mean when is

known and unknown.


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CONFIDENCE INTERVAL FOR

MEAN


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CONFIDENCE INTERVAL FOR THE

MEAN

2 2, X z X z

n nE E

W W

2 2

,s s

X z X z n n

E E

, 1 , 12 2,

n n

s s X t X t

n nE E

The ( 1 ) 100 % confidence interval for


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T- DISTRIBUTIONS

The number of values that are free

to vary after a sample statistic has

been computed


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ROUNDING RULE

When you are computing a confidenceinterval for a population mean by usingraw data, round off to one more decimal

place than the number of decimal placesin the original data.

When you are computing a confidence

interval for a population mean by using asample mean and standard deviation,round off to the same number of decimalplaces as given for the mean.


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EXAMPLE 1

The mass of vitamin E in a capsulemanufactured by a certain drug companyis normally distributed with standard

deviation 0.042 mg.Arandom sample of 5 capsules wasanalyzed and the mean mass of vitamin Ewas found to be 5.12 mg.

Find the 95% confidence interval for thepopulation mean mass of vitamin E percapsule.


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EXAMPLE 2

Aplant produces steel sheets whose

weights are known to be normally

distributed with a standard deviation of2.4 kg.

Arandom sample of 36 sheets had a

mean weight of 3.14 kg.

Find the 99% confidence interval for thepopulation mean weight.


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EXAMPLE 3

Arandom number of 100 pieces of wood

are cut using a machine.

The sample mean of length in cm is 1.06cm and the standard deviation is 0.08

cm.

Find the 95% confidence interval for

mean length all the woods cut by themachine.

What is the width of this confidence

interval?


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EXAMPLE 4

The mean IQ score for 25 UMP students is

115 with standard deviation 10.

If the IQ score for all UMP students isnormally distributed.

Find the 95% confidence interval for the

mean IQ score for all UMP students.


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EXAMPLE 5

The result Xof a stress test is known to be

normally distributed random variable

with mean and standard deviation 1.3.

It is required to have a 95% confidence

interval for with total width less than 2.

Find the least number of tests that should

be carried out to achieve this.


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EXAMPLE 6

8 UMP students are randomly chosen and

the value of their CPAhas been collected

as below.

3.20 2.76 2.94 3.41

2.92 2.99 3.01 3.11

Find the 95% confidence interval for theCPAmean for all UMP students.


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EXAMPLE 7

The heights of men in a particular districtare distributed with mean cm and thestandard deviation cm. On the basis ofthe results obtained from a randomsample of 100 men from the district, the95% confidence interval for wascalculated and found to be (177.22 cm ,

179.18 cm). Calculate the value of samplemean and standard deviation.


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EXAMPLE 8

A90% confidence interval for apopulation mean based on 144observations is computed to be (2.7, 3.4).

How many observations must be made sothat a 90% confidence interval willspecify the mean to within 0.2?


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3.43.4 CONFIDENCE INTERVAL FOR THE

DIFFERENCE BETWEEN 2 MEAN



1. Find the confidence interval for the difference between

two means when s are known.


two means when s are unknown and equal.


two means when s are unknown and not equal.


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CONFIDENCE INTERVAL FOR THE

DIFFERENCE BETWEEN 2 MEAN


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EXAMPLE 1

The mean of sleep time for 50 IPTSstudents are 7 hours with standarddeviation of 1 hour.

The mean of sleep time for 60 IPTAstudents is 6 hours with standard deviationof 0.7 hour.

Find the 99% confidence interval for thedifferent mean of sleep time between theIPTS and IPTAstudents.

1. Assume the population variance are same

2. Assume the population variance are different


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EXAMPLE 2

The mean of sleep time for 20 IPTSstudents are 7 hours with standarddeviation of 1 hour.

The mean of sleep time for 15 IPTAstudents is 6 hours with standard deviationof 0.7 hour.

Find the 99% confidence interval for thedifferent mean of sleep time between the

IPTS and IPTAstudents.

1. Assume the population variance are same

2. Assume the population variance are different


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EXAMPLE 3

Find the 95% confidence interval for the

different mean of childrens sleep time and

adults sleep time if given that the

variances for childrens sleep time is 0.81

hours while for adults is 0.25 hours.

The mean sample sleep time for 30

childrens are 10 hours while for 40 adultsare 7 hours.


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EXAMPLE 4

Two groups of students are given a problemsolving test, and the results are compared. Thedata are follows:

Mathematics Majors ComputerScience majors

Find the 98% confidence interval for the differentmean of test marks between the two groups ofstudents. Assume the variance population testmarks are same for both groups.

1

1

1

29

83.6

3.3

n

x

s

!

!

!

2

2

1

28

79.2

2.8

n

x

s

!

!

!


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EXAMPLE 5Amedical researcher wishes to see whether thepulse rates of nonsmokers are lower than thepulse rates of smokers. Samples of 110 smokersand 120 nonsmokers are selected. The resultsare shown here.

Smokers Nonsmokers

Find the 90% confidence interval for thedifferent mean between pulse rates ofnonsmokers and the pulse rates of smokers.Assume that the variance pulse rates for bothpopulations are not same.

1

1

1

90

5

110

X

s

n

!

!

!

2

2

2

88

6

120

X

s

n

!

!

!


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3.53.5 CONFIDENCE

INTERVAL

FOR THE PROPORTION



1. Find the confidence interval for a proportion


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The ( 1 ) 100 % confidence interval for proportionp

0

2 0 0

,

1

test

p ppqp z z

np p

n

E

s !

)) ))

where

and 1x

p q pn

! !

) ) )


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EXAMPLE 1

1. 23 from 100 families in a village are poor. Findthe 99% confidence interval poorness rate forthis village.

2. Asurvey was undertaken of the use of theinternet by residents in a large city and it wasdiscovered that in a random sample of 150residents, 45 logged on to the internet at leastonce a day. Calculate an approximate 90%

confidence interval forp, the proportion ofresidents in the city that log on to the internetat least once a day.


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3.63.6 CONFIDENCE INTERVAL

DDIFFERENCEIFFERENCE BETWEENBETWEEN

22 PPROPORTIONSROPORTIONS



1. Find the confidence interval for the difference betweentwo proportions.


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The (

1 )

100% con

fidence

interval

for the d

ifferent

proportionsp1 p2

1 1 2 21 22

1 2

p q p qp p z

n nE

s

) ) ) )) )


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EXAMPLES

1. Given .Find the 95% confidence intervals for.

2. In a sample of 200 surgeons, 15% thoughtthe government should control healthcare. In a sample of 200 general

practitioners, 21% felt the same way. Find95% confidence interval for the differenceof proportions for surgeons andpractitioners.

1 1 2 20.6, 75, 0.3, 100p n p n! ! ! !

) )

1 2p p


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FORVARIANCES AND

STANDARD DEVIATIONS

OBJECTIVESOBJECTIVES After completing this chapter, you should be able to

1. Find the confidence interval for a variance and a

standard deviation.


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The ( 1 ) 100 % confidence interval for the variance

2 22 2

, 1 1 , 12 2

1 1,

n n

n s n s

E EG G

2W

2 22 2

, 1 1 , 12 2

1 1,

n n

n s n s

E EG G

The ( 1 ) 100 % confidence interval for the standard deviation W

Where 2 2

12

1~

n

n sG

W

(Chi-square distribution)


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EXAMPLE 1

Arandom sample of 10 rulers produce by

a machine gives a group of data below, in

cm.

100.13, 100.07, 100.02, 99.99, 99.88,

100.14, 100.03, 100.10, 99.92, 100.21

Find the 95% confidence interval for theheight variance and standard deviation of

all the rulers produce by the machine.


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EXAMPLE 2

A factory has a machine thats designed tofilled boxes with an average of 24 ounces ofcereal, and the population standard deviationfor this filling process is expected to be 0.1ounce.

Thus, if the machine is working properly, thepopulation variance should be 0.01 squaredounce.To estimate the value of population variance,an employee selected a random sample of 15boxes from a supply filled by the machine and

found that the sample variance was 0.008squared ounce.Whats the 95% confidence interval for thepopulation variance and standard deviation?


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FOR 2 VARIANCES AND

STANDARD DEVIATIONS



1. Find the confidence interval for the confidence interval

for the variance and a standard deviation proportion.


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The ( 1 ) 100 % confidence interval

for the variance proportion

Where

2

1

2

2

W

W

2 1

1 2

2 2

1 1

2 2 , 1, 12

2 2, 1, 12

1 ,n n

n n

s s F s F s

E

E

1 2

2 2

1 1

1, 12 2

2 2

n ns Fs

W

W (F distribution)


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EXAMPLE 1

The machined in example 1 (topic 2.7) isserviced. Arandom sample of 12 rulersproduces by the machine after the servicedmade give a group of data below.

100.03, 100.01, 100.02, 100.04,

99.90, 99.96, 100.04, 100.06,

100.08, 99.98, 100.11, 100.05

Find the 95% confidence interval forvariance proportion for all rulers producesby the machine before and after the service.


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EXAMPLE 2

Before service, a machine can packed10 packets of sugar with varianceweight 64 g while after service thevariance weight for 5 packets of sugar

are 25 g . Find the 99% confidenceinterval for variance proportion for allsugar produces by the machine beforeand after the service.


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CONCLUSION

An important aspect of inferential statistics isestimation

Estimations of parameters of populations are

accomplished by selecting a random sample fromthat population and choosing and computing astatistic that is the best estimator of theparameter

Statisticians prefer to use the interval estimaterather than point estimate because they can be95%,99% or else confidence that their estimatecontains the true parameter and also determinethe minimum sample size necessary.


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Chapter 3 - Sampling Distribution and Confidence Interval1