Matakuliah
Tahun
Versi
: I0262 – Statistik Probabilitas
: 2007
: Revisi
Pertemuan 07
Pendugaan Parameter
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa akan dapat menghasilkan
dugaan parameter, nilai tengah, proporsi
dan ragam.
2
Outline Materi
• Pendugaan Titik
• Pendugaan Selang : nilai tengah, proporsi
dan ragam.
3
Point Estimation
• In point estimation we use the data from the
sample to compute a value of a sample statistic
that serves as an estimate of a population
parameter.
• We refer to x as the point estimator of the
population mean .
p the point estimator of the population
• s is
standard deviation .
•
is the point estimator of the population
proportion p.
4
Sampling Error
• The absolute difference between an unbiased
point estimate and the corresponding population
parameter is called the sampling error.
• Sampling error is the result of using a subset of
the population (the sample), and not the entire
population to develop estimates.
• The sampling errors are:
| x   | for sample mean
| s -  | for sample standard deviation
| p  p | for sample proportion
5
Interval Estimation
• Interval Estimation of a Population Mean:
Large-Sample Case
• Interval Estimation of a Population Mean:
Small-Sample Case
• Determining the Sample Size
• Interval Estimation of a Populationx

[--------------------- x ---------------------]
Proportion
[--------------------- x ---------------------]
[--------------------- x ---------------------]
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Interval Estimate of a Population
Mean:
Large-Sample Case (n > 30)
• With  Known
x  z /2
where:

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
7
Interval Estimate of a Population Mean:
Large-Sample Case (n > 30)
• 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.
s
x  z /2
n
8
Interval Estimation of a Population
Mean:
Small-Sample Case (n < 30) with 
Unknown
• Interval Estimate
x  t /2
s
n
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
where
9
Interval Estimation
of a Population Proportion
• Interval Estimate
p  z / 2
where:
p (1  p )
n
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
p is the sample proportion
10
Point Estimator of the Difference
Between
the Means of Two Populations
• Let 1 equal the mean of population 1 and 2 equal
the mean of population 2.
• The difference between the two population means is
1 - 2.
• To estimate 1 - 2, we will select a simple random
sample of size n1 from population 1 and a simple
random sample of size n2 from population 2.
• Let x1 equal the mean of sample 1 and x2 equal the
mean of sample 2.
• The point estimator of the difference between the
means of the populations 1 and 2 is x1  x2 .
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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  z / 2  x1  x2
where:
1 -  is the confidence coefficient
• Interval Estimate with 1 and 2 Unknown
x1  x2  z / 2 sx1  x2
where:
sx1  x2
s12 s22


n1 n2
12
Interval Estimate of 1 - 2:
Small-Sample Case (n1 < 30
and/or n2 < 30)
• Interval Estimate with  2 Known
x1  x2  z / 2  x1  x2
where:
 x1  x2
1 1
  (  )
n1 n2
2
13
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
1 1
 s (  )
n1 n2
2
2
2
(
n

1
)
s

(
n

1
)
s
1
2
2
s2  1
n1  n2  2
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• Selamat Belajar Semoga Sukses.
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