Point and Interval Estimation-26!08!2011

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Point and Interval Estimation


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Estimation

The objective of estimation is to determinethe approximate value of a populationparameter on the basis of a sample statistic.

For example, sample mean is employed toestimate the population mean. We refer tothe sample mean as the estimator of the

population mean.

Once the sample mean has been computed,its value is called the estimate.


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Point Estimation

We can use sample data to estimate a populationparameter in two ways.

Point Estimation and Interval Estimation

In point estimation procedure, we make an attemptto compute a numerical value, from sampleobservation, which could be taken as an

approximate to the parameter.

Sample mean is the best possible estimatorof the population mean , because it is unbiased,

consistent, and relatively efficient.

n

XX


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Evaluating the Goodness of a

Point Estimator

Unbiasedness

An unbiased estimator of a population parameteris an estimator whose expected value is equal tothat parameter.

Sample mean is an unbiased estimator of thepopulation mean because

If we define sample variance using n in thedenominator, the resulting statistic would be a

biased estimator of the population variance.But if, then

So now is an unbiased estimator of thepopulation variance

)(XE

1

)( 22

n

XXS

22 )( SE

2S

2


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Point Estimator

Consistency

An unbiased estimator is said to be consistent, ifthe difference between the estimator and the

population parameter becomes smaller as thesample size grows larger. is a consistentestimator of, because the variance of is

This implies that as n grows larger, the variance ofgrows smaller. As a consequence, larger

sample size will results better estimate.

X

X ./2 n

X


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Point Estimator

Relative Efficiency

If there are two unbiased estimators of aparameter, the one whose variance is smaller issaid to be relatively efficient.

Sample mean is an unbiased estimator of thepopulation mean and its variance is

Statisticians have established that (sampling from

a normal population) the sample median is also anunbiased estimator of the population mean, but itsvariance is 1.57 Consequently, we say thatthe sample mean is relatively more efficient than

the sample median.

./2 n

./2 n


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Interval Estimation

We have to be extremely lucky to have a sample which has

a mean exactly equal to the population mean, otherwise itwill be a little higher or a little lower.

We, therefore, try to determine two values, instead of one

point estimate, within which the true value of the parameteris expected to fall.

We can also attached a certain degree of confidence to ourestimated value. The two values which are expected tocontain the true value of a population parameter are calledthe confidence limits (the lower confidence limit (LCL) andthe upper confidence limit (UCL) and the two together arecalled confidence intervals.


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Confidence Interval Estimate of

Mean (Large Sample)

1

1

1

22

22

22/

nZX

nZXP

nZXnZP

Z

n

XZP

1

1

22

22

nZX

nZXP

n

ZX

n

ZXP

nZX

2

nZX

2

Upper Confidence Limit (UCL) =

Lower Confidence Limit (UCL) =

122

ZZZP


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Mean (Large Sample)

90% confidence limits are

nXand

nX

64.164.1


nX

nX

96.1,96.1

99% confidence limitsare

nXnX

575.2,575.2 is usually not known, therefore, for large samples it can be

replaced by s (the sample standard deviation) which may be

calculated from the sample by using the following the formula

1

)( 22

n

XXS


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Example

A random sample of 64 students in a class

made an average score of 60, with a

standard deviation of 15. Construct 99%confidence interval estimate for the mean

score of entire class.

=> Also construct 90% and 95 % confidence

interval estimate for the mean score of entire

class.


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Example- Solution

elyapproximat

n

XitconfidenceUpper

theAlso

elyapproximat

nXitconfidenceLower

nXervalConfidence

65

83.64

8

1558.260

58.2lim

55

16.55

8

1558.260

58.2lim

58.2int%99


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Example- Solution

675.6308.63325.56925.56815

\/64

%95#%9060

UCLUCL

LCLLCLnS

n

X


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Mean (Small Sample)

In practice is usually notknown, also the application of central limittheorem is feasible only when sample size is

large. We thus need to know some suchdistribution, based on random samples,where we could overcome these difficulties.

So if population S.D is known and sample

size is large, (n = 30 is considered as large)use Z distribution. If population S.D is notknown, but sample size is large, replace

by s (sample S.D) and use Z distribution

)..( populationofDS


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Mean (Small Sample)

But if sample size is small (n


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Definition

Degrees of Freedom (df) = n - 1

Any

#Specific

#

so that x = 80

Any

#

n = 10 df= 10 - 1 = 9

Any

#Any

#Any

#Any

#Any

#Any

#Any

#


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Important Properties of the Student t

Distribution

1. The Student t distribution is different for different sample sizes (seeFigure 6-5 for the cases n = 3 and n = 12).

2. The Student t distribution has the same general symmetric bellshape as the normal distribution but it reflects the greatervariability (with wider distributions) that is expected with smallsamples.

3. The Student tdistribution has a mean oft = 0 (just as the standardnormal distribution has a mean ofz = 0).

4. The standard deviation of the Student t distribution varies with the

sample size and is greater than 1 (unlike the standard normaldistribution, which has a = 1).

5. As the sample size n gets larger, the Student t distribution getscloser to the normal distribution. For values ofn = 30, thedifferences are so small that we can use the critical z valuesinstead of developing a much larger table of critical t values.


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Student t

distribution

with n = 3

Student t Distributions for

n = 3 and n = 12

0

Student t

distributionwith n = 12

Standard

normal

distribution

Figure 6-5


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Mean (Small Sample)

ExampleA random sample of 10 packets wastaken and is found to have a mean

weight of 60 grams and a standarddeviation of 12 grams. What is the meanweight of the population

(a) with 95% confidence?

(b) with 99% confidence?


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Mean (Small Sample)


confidence

dfSX

%95

91101260

025.02

05.0 n

stX

58.68~42.51262.2025.0,9 t

33.7267.47

33.1260250.3

10

12250.360

005.001.0%99

005.0,92

and

tt


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Inference about a Population

Variance

2

.

1

)( 22

n

XXS

2S

2

Point Estimator

The point estimator for is the sample variance

is an unbiased, consistent estimator of


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Sampling distribution ofTo create the sampling distribution of the sample variance, we

repeatedly take samples of size n from a normal populationwhose variance is , calculate for each sample

2S

2

2S

22

2

2

)()1(

1

)(

XXSn

n

XXS

Mathematician have shown that the sum of squared difference

2)( XX 2)1( Sntoequaliswhich

divided by the population variance is distributed according

to what is called the chi-squared distribution provided that the

population is normal.


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wheren = sample size

s 2= sample variance

2 = population variance

Chi-Square Distribution

X2= 2

(n - 1) s2


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Properties of the Distribution of

the Chi-Square Statistic

1. The chi-square distribution is not symmetric, unlikethe normal and Student t distributions.

As the number of degrees of freedom increases, thedistribution becomes more symmetric. (continued)

0 5 10 15 20 25 30 35 40 45

Figure 6-8 Chi-Square Distribution fordf= 10

and df= 20

df= 10

df= 20

Figure 6-7 Chi-Square Distribution

All values are nonnegative

Not symmetric

x2

0

P i f h Di ib i f


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Properties of the Distribution of

the Chi-Square Statistic(continued)

2. The values of chi-square can be zero or positive, butthey cannot be negative.

3. The chi-square distribution is different for eachnumber of degrees of freedom, which is df= n - 1in this section. As the number increases, the chi-square distribution approaches a normal

distribution.


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A

A

c2Ac21-A

The c2 table

Degrees offreedom

1 0.0000393 0.0001571 0.0009821 . . 6.6349 7.87944..

10 2.15585 2.55821 3.24697 . . 23.2093 25.1882. . . . . .. . . . . . . .

c2.995c2.990c

2.975c

2.010c

2.005

.990 .010

=.01

=.01

1 - A =.99A

c2.01,1023.2093

I f b t P l ti


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Variance

1)1(

1

lim%100)1(

2

2

2

2

2

12

2

22

21

2

XSn

XP

XXXP

areitsconfidenceThe

1)1()1(

1)1()1(

1)1(

1

)1(

21

2

22

2

2

2

2

2

22

21

2

2

2

2

2

22

21

2

X

Sn

X

SnP

X

Sn

X

SnP

Sn

X

Sn

XP

I f b t P l ti


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VarianceThe %100)1( lower and upper confidence limits of the

population variance are

21

2

2

2

2

2

)1(lim

)1(lim

XSnitconfidenceUpper

X

SnitconfidenceLower

2

2X

21

2X

Where n-1 are the degrees of freedom and the value of

and

are available from the chi-square table against

(n-1) degrees of freedom and the appropriate level of significance.

95.02

05.0205.0

21.0 XXIf

areatheofcontainwillXX %95025.0205.0 975.02

025.02

I f b P l i


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Variance

Example

A random sample of 10 bottles of a cough syrup

found to have an average alcohol content of 3.5

m.l. with a variance of 0.64 m.l. Construct a 99percent confidence interval for the true standard

deviation alcohol contents of the cough syrup.

Point and Interval Estimation-26!08!2011

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