Matakuliah
Tahun
: I0272 - STATISTIK PROBABILITAS
: 2009
PENDUGAAN PARAMETER
Pertemuan 8
Materi
• Pendugaan parameter proporsi (pendugaan titik
dan selang)
• Pendugaan parameter ragam (pendugaan titik
dan selang)
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3
Estimasi Interval 1 Proporsi Populasi
Selang kepercayaan bagi p untuk contoh berukuran besar :
Bila p̂ adalah proporsi keberhasilan dalam suatu contoh acak
berukuran n, dan q̂ = 1 - p̂ ,maka selang kepercayaan (1 )100% bagi p adalah:
Pˆ  z
2
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pˆ qˆ
 p  pˆ  z
n
2
pq
n
44
Estimasi Beda 2 Proporsi
Selang kepercayaan bagi p1 – p2 untuk contoh berukuran
besar:
Bila pˆ1 dan pˆ 2 masing-2 adalah proporsi keberhasilan dalam
contoh acak berukuran n1 dan n2, serta qˆ1  1  pˆ1 dan qˆ2  1  pˆ 2
, maka selang kepercayaan (1 - ) 100% bagi p1 – p2
adalah :
 pˆ1  pˆ 2   z
2
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pˆ1qˆ1 pˆ 2qˆ2

 p1  p2   pˆ1  pˆ 2   z
n1
n2
2
pˆ1qˆ1 pˆ 2qˆ2

n1
n2
5
Inferences About Population Variances
• Inference about a Population Variance
• Inferences about the Variances of Two Populations
•
•
Chi-Square Distribution
Interval Estimation of
2
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66
Chi-Square Distribution


We will use the notation  to denote the value for
the chi-square distribution that provides an area of 
to the right of the stated 2 value.
2
2
2
.975
  2  .025
For example, there is a .95 probability of obtaining a
2 (chi-square) value such that
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7
Interval Estimation of 2
2
.975

95% of the
possible 2 values
.025  2
.975
(n  1)s 2
2
2
 .025
2
 .025
.025
2
0
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8
Interval Estimation of 2

There is a (1 – ) probability of obtaining a 2 value
such that
2
2
2
(1 / 2)     / 2

Substituting (n – 1)s2/2 for the 2 we get
 (12  / 2) 

(n  1) s 2
2
 2 / 2
Performing algebraic manipulation we get
( n  1) s 2
 2 / 2
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 2 
( n  1) s 2
 2(1 / 2)
9
Interval Estimation of
2
Interval Estimate of a Population Variance
( n  1) s 2
 2 / 2
 2 
( n  1) s 2
 2(1 / 2)
where the  values are based on a chi-square distribution
with n - 1 degrees of freedom and where 1 -  is the
confidence coefficient.
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10
Interval Estimation of
• Interval Estimate of a Population Standard Deviation
Taking the square root of the upper and lower
limits of the variance interval provides the confidence
interval for the population standard deviation.
(n  1) s 2
(n  1) s 2
 
2
 / 2
 (12  / 2)
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11
Interval Estimation of

2
Example: Buyer’s Digest (A)
Buyer’s Digest rates thermostats manufactured for home
temperature control. In a recent test, 10 thermostats
manufactured by ThermoRite were selected and placed
in a test room that was maintained at a temperature of
68oF.
The temperature readings of the ten thermostats are
shown on the next slide.
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12
Interval Estimation of 2

Example: Buyer’s Digest (A)
We will use the 10 readings below to
develop a 95% confidence interval
estimate of the population variance.
Thermostat
1
2
3
4
5
6
7
8
9
10
Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2
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13
For n - 1 = 10 - 1 = 9 d.f. and  = .05
Selected Values from the Chi-Square Distribution Table
Area in Upper Tail
Degrees
of Freedom
5
6
7
8
9
10
.99
0.554
0.872
1.239
1.647
2.088
.975
0.831
1.237
1.690
2.180
2.700
.95
1.145
1.635
2.167
2.733
3.325
.90
1.610
2.204
2.833
3.490
4.168
2.558 3.247
3.940
4.865 15.987 18.307 20.483 23.209
Our
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.10
9.236
10.645
12.017
13.362
14.684
.05
11.070
12.592
14.067
15.507
16.919
.025
12.832
14.449
16.013
17.535
19.023
.01
15.086
16.812
18.475
20.090
21.666
2
value
.975
14
For n - 1 = 10 - 1 = 9 d.f. and  = .05
2.700 
.025
(n  1)s 2
2
2
  .025
Area in
Upper Tail
= .975
2
0 2.700
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15
Interval Estimation of
2
For n - 1 = 10 - 1 = 9 d.f. and  = .05
Selected Values from the Chi-Square Distribution Table
Area in Upper Tail
Degrees
of Freedom
5
6
7
8
9
10
.10
9.236
10.645
12.017
13.362
14.684
.05
11.070
12.592
14.067
15.507
16.919
.025
12.832
14.449
16.013
17.535
19.023
.01
15.086
16.812
18.475
20.090
21.666
.975
0.831
1.237
1.690
2.180
2.700
.95
1.145
1.635
2.167
2.733
3.325
.90
1.610
2.204
2.833
3.490
4.168
2.558 3.247
3.940
4.865 15.987 18.307 20.483 23.209
.99
0.554
0.872
1.239
1.647
2.088
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Our
2
.025
value
16
Interval Estimation of
2
n - 1 = 10 - 1 = 9 degrees of freedom and  =
.05
2.700 
(n  1)s 2

2
 19.023
.025
Area in Upper
Tail = .025
0
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2
2.700
19.023
17
Interval Estimation of 2
• Sample variance s2 provides a point estimate of

2.
A 95% confidence interval for the population variance
is given by:
(10  1). 70
(10  1). 70
 2 
19. 02
2. 70
.33 < 2 < 2.33
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18
Pendugaan Rasio Dua Varians
Selang kepercayaan bagi
 12
 22
• Bila S12 dan S22 adalah ragam dua contoh bebas
berukuran n1 dan n2 yang diambil dari populasi
 12
normal, maka selang kepercayaan (1 - ) 100% bagi 2
2
.
s12
 12
s12
1
 2  2 f  v2 ,v1 
2
s 2 f  v1 ,v2   2
s2 2
2
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1919