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7/30/2019 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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Evaluating the Goodness of a
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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Evaluating the Goodness of a
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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Confidence Interval Estimate of
Mean (Large Sample)
90% confidence limits are
nXand
nX
64.164.1
95% confidence limits are
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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Confidence Interval Estimate of
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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Confidence Interval Estimate of
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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Confidence Interval Estimate of
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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Confidence Interval Estimate of
Mean (Small Sample)
95% confidence limits are
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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Inference about a Population
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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Inference about a Population
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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Inference about a Population
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.