Matakuliah
Tahun
: I0272 - STATISTIK PROBABILITAS
: 2009
PENDUGAAN PARAMETER
Pertemuan 7
Materi
• Pendugaan Titik dan Selang
• Pendugaan Selang: Nilai tengah dan beda dua
nilai tengah
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 Pendugaan Titik dan Selang
• Pendugaan Parameter. Suatu statistik merupakan nilai
dugaan bagi parameter populasi . Misalnya x
merupakan nilai dugaan bagi , penduga ini disebut
Penduga Titik.
• Definisi : Suatu statistik disebut penduga tak bias bagi
parameter  bila  = E() = 
• Penduga yang lebih baik: Dugaan Selang. Secara
umum: dugaan selang bagi parameter populasi 
ˆ 
ˆ
adalah suatu yang berbentuk 
1
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4
 Pendugaan Selang: Nilai tengah dan beda 2 nilai tengah
• Interval Estimation of a Population Mean: 1 populasi
– Large-Sample Case (n > 30)
– Small-Sample Case (n < 30)
• Interval Estimation of a Population Mean: 2 populasi
– Large-Sample Case (n > 30)
– Small-Sample Case (n < 30)

x
[--------------------- x ---------------------]
[--------------------- x ---------------------]
[--------------------- x ---------------------]
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Interval Estimate of a Population Mean:
Large-Sample Case (n > 30)
• With  Known
where:
x  z /2

n
x
is the sample mean
1 - is the confidence coefficient
z/2 is the z value providing an area of
/2 in the upper tail of the standard
normal probability distribution

is the population standard deviation
n
is the sample size
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• With  Unknown
In most applications the value of the population standard deviation is
unknown. We simply use the value of the sample standard
deviation, s, as the point estimate of the population standard
deviation.
x  z /2
s
n
Small-Sample Case (n < 30) with  Unknown
Interval Estimate:
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Interval Estimation of a Population Mean:
Small-Sample Case (n < 30)
• Population is Not Normally Distributed. The only option
is to increase the sample size to n > 30 and use the
large-sample interval-estimation procedures.
• Population is Normally Distributed and
is Known. The
large-sample interval-estimation procedure can be
used.
• Population is Normally Distributed and
is Unknown.
The appropriate interval estimate is based on a
probability distribution known as the t distribution.
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Small-Sample Case (n < 30) with  Unknown
• Interval Estimate
x  t /2
s
n
where 1 - = the confidence coefficient
t /2 = the t value providing an area of /2
in the upper tail of a t distribution
with n - 1 degrees of freedom
s = the sample standard deviation
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Interval Estimate of 1 - 2:
Large-Sample Case (n1 > 30 and n2 > 30)
• Interval Estimate with 1 and 2 Known
x1  x2
where:
1 - is the confidence coefficient
• Interval Estimate with 1 and 2 Unknown
 z / 2  x1  x2
x1  x2  z / 2 sx1  x2
where:
sx1  x2
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s12 s22


n1 n2
10
Interval Estimate of 1 - 2:
Small-Sample Case (n1 < 30 and/or n2 < 30)
• Interval Estimate with  2 Known
where:
x1  x2  z /2 x1  x2
 x1  x2
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1 1
  (  )
n1 n2
2
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Interval Estimate of 1 - 2:
Small-Sample Case (n1 < 30 and/or n2 < 30)
• Interval Estimate with  2 Unknown
x1  x2  t /2 sx1  x2
where:
sx1  x2
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1 1
 s (  )
n1 n2
2
s 
2
2
( n1  1) s1
2
 ( n2  1) s2
n1  n2  2
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Contoh Soal: Specific Motors
• Point Estimate of the Difference Between 2 Population
Means
= mean miles-per-gallon for the population
of M cars
2 = mean miles-per-gallon for the population
of J cars
1
Point estimate of
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1
-
x  x2 = 29.8 - 27.3 = 2.5 mpg.
2= 1
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Contoh Soal: Specific Motors
•
95% Confidence Interval Estimate of the Difference Between Two Population
Means: Small-Sample Case
2
2
2
2
(
n

1
)
s

(
n

1
)
s
11
(
2
.
56
)

7
(
1
.
81
)
1
2
2
s2  1

 5. 28
n1  n2  2
12  8  2
x1  x2  t.025
1 1
1 1
s (  )  2. 5  2.101 5. 28(  )
n1 n2
12 8
2
= 2.5 + 2.2 or .3 to 4.7 miles per gallon.
We are 95% confident that the difference between the
mean mpg ratings of the two car types is from .3 to 4.7 mpg (with the M car
having the higher mpg).
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