Matakuliah
: I0014 / Biostatistika
Tahun
: 2008
Pendugaan Parameter (II)
Pertemuan 10
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa dapat menghitung pendugaan nilai
tengah populasi (C3)
• Mahasiswa dapat menghitung pendugaan ragam
populasi (C3)
• Mahasiswa dapat menghitung pendugaan proporsi
populasi (C3)
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Outline Materi
• Pendugaan Nilai tengah
(  dan 1  2 )
• Pendugaan Ragam
(
2
dan  /  )
• Pendugaan Proporsi
( p dan p1  p2 )
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2
1
2
2
Jenis Penduga
•
•
Point Estimate
– A single-valued estimate.
– A single element chosen from a sampling distribution.
– Conveys little information about the actual value of the
population parameter, about the accuracy of the estimate.
Confidence Interval or Interval Estimate
– An interval or range of values believed to include the
unknown population parameter.
– Associated with the interval is a measure of the
confidence we have that the interval does indeed
contain the parameter of interest.
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Selang Kepercayaan (1- )100%
We define z as the z value that cuts off a right-tail area of  under the standard
2
2
normal curve. (1-) is called the confidence coefficient.  is called the error
probability, and (1-)100% is called the confidence level.
S tand ard Norm al Distrib ution
0.4
(1   )
(1- )100% Confidence Interval:
f(z)
0.3
x  z
0.2
0.1


2
2
2
0.0
-5
-4
-3
-2
-1
z 
2
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0
1
Z
z
2
2
3
4
5

n
Selang Kepercayaan untuk 
bila  Tidak Diketahui
A (1-)100% confidence interval for  when  is not known
(assuming a normally distributed population):
x  t
2
s
n
where t is the value of the t distribution with n-1 degrees of
2

freedom that cuts off a tail area of 2 to its right.
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Penduga Selang untuk Proporsi
A large - sample (1 -  )100% confidence interval for the population proportion, p:
pq

p  z
 n
2
where the sample proportion, p,
 is equal to the number of successes in the sample, x ,
divided by the number of trials (the sample size), n , and q = 1 - p.

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Selang Kepercayaan untuk
Ragam
A (1-)100% confidence interval for the population variance * (where the
population is assumed normal):

2
2
 ( n  1) s , ( n  1) s 
  2
2  
1


2
2
where   is the value of the chi-square distribution with n-1 degrees of freedom
2
2
that cuts off an area
cuts off an area of

2
2
2

to its right and

1
2
is the value of the distribution that
to its left (equivalently, an area of 1 

2
to its right).
* Note: Because the chi-square distribution is skewed, the confidence interval for the
population variance is not symmetric
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Selang Kepercayaan untuk
Beda Dua Mean Populasi
A large-sample (1-)100% confidence interval for the difference
between two population means, 1- 2 , using independent random
samples:
( x1  x 2 )  z

2
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2
2
s1
s2

n1 n 2
• When sample sizes are small (n1< 30 or n2< 30 or both),
and both populations are normally distributed, the test
statistic
( x  x )  (   2 )0
t  1 2 2 12
s1 s2

n1 n2
• has approximately a t distribution with degrees of freedom
given by (round downward
2 to the
2 2 nearest integer if
 s1 s2 
necessary):
 n1  n2 

df  2 2
2
2
 s1 
 s2 
 
 
n
 1
 n2 

n1  1 n2  1
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Pendugaan Ragam Gabungan
A pooled estimate of the common population variance, based on a sample
variance s12 from a sample of size n1 and a sample variance s22 from a sample of
size n2 is given by:
2
2
(
n

1
)
s

(
n

1
)
s
1
2
2
s2p  1
n1  n2  2
The degrees of freedom associated with this estimator is:
df = (n1+ n2-2)
The pooled estimate of the variance is a weighted average of the two individual
sample variances, with weights proportional to the sizes of the two samples. That
is, larger weight is given to the variance from the larger sample.
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Selang Kepercayaan
menggunakan Ragam
Gabungan
A (1-) 100% confidence interval for the difference between two
population means, 1- 2 , using independent random samples and
assuming equal population variances:
( x1  x2 )  t

2
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2 1
sp 
 n1



n2 
1
Selang Kepercayaan Beda Dua
Proporsi
A (1-) 100% large-sample confidence interval for the difference
between two population proportions:
( p1  p 2 )  z

2
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 p (1  p )
1
 1
 n1 

p (1  p ) 
2
2 
n2


Selang kepercayaan Rasio Dua
Ragam
2
1
A (1 -  ) 100% confidence interval for 2 :
2
 s12
 s2
2
 F





,
F
 1  


2
s1
2
s2
where F is the value obtained through the table and F
 1- 
is the left - tailed value of the distribution
obtained as the reciprocal of the F value with reversed - order degrees of freedom.
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Penutup
• Sampai saat ini Anda telah mempelajari
pendugaan titik dan selang, baik untuk satu
populasi maupun dua populasi
• Untuk dapat lebih memahami penggunaan
pendugaan tersebut, cobalah Anda pelajari materi
penunjang, dan mengerjakan latihan
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