Matakuliah
Tahun
Versi
: I0134 – Metoda Statistika
: 2005
: Revisi
Pertemuan 16
Pendugaan Parameter
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa dapat menghitung penduga
selang dari rataan, proporsi dan varians.
2
Outline Materi
•
•
•
•
Selang nilai tengah (rataan)
Selang beda nilai tengah (rataan)
Selang proporsi dan beda proporsi
Selang varians dan proporsi varians
3
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 ---------------------]
4
Interval Estimation of a Population
Mean:
Large-Sample Case
• Sampling Error
• Probability Statements about the Sampling
Error
• Constructing an Interval Estimate:
Large-Sample Case with  Known
• Calculating an Interval Estimate:
Large-Sample Case with  Unknown
5
Sampling Error
• The absolute value of the difference
between an unbiased point estimate
and the population parameter it
estimates is called the sampling error.
• For the case of a sample mean
estimating a population mean, the
sampling error is
Sampling Error = | x  |
6
Interval Estimate of a Population Mean:
Large-Sample Case (n > 30)
• With  Known
x  z /2
where:
1 -
x
z/2

n

n
is the sample mean
is the confidence coefficient
is the z value providing an area of
/2 in the upper tail of the standard
normal probability distribution
is the population standard deviation
is the sample size
7
Interval Estimate of a Population
Mean:
Large-Sample Case (n > 30)
x  z /2
s
n
• 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.
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 9
where
Contoh Soal: Apartment Rents
• Interval Estimation of a Population Mean:
Small-Sample Case (n < 30) with  Unknown
A reporter for a student newspaper is writing
an
article on the cost of off-campus housing. A
sample of 10 one-bedroom units within a half-mile
of campus resulted in a sample mean of $550 per
month and a sample standard deviation of $60.
Let us provide a 95% confidence interval
estimate of the mean rent per month for the
population of one-bedroom units within a halfmile of campus. We’ll assume this population to
be normally distributed.
10
Contoh Soal: Apartment Rents
• t Value
At 95% confidence, 1 -  = .95,  = .05, and /2 =
.025.
t.025 is based on n - 1 = 10 - 1 = 9 degrees of freedom.
In the t distribution table we see that t.025 = 2.262.
Degrees
Area in Upper Tail
of Freedom
.10
.05
.025
.01
.005
.
.
.
.
.
.
7
1.415
1.895
2.365
2.998
3.499
8
1.397
1.860
2.306
2.896
3.355
9
1.383
1.833
2.262
2.821
3.250
10
1.372
1.812
2.228
2.764
3.169
.
.
.
.
.
.
11
Estimation of the Difference Between the
Means
of Two Populations: Independent Samples
• Point Estimator of the Difference between
the Means of Two Populations
• Sampling Distribution x1  x2
• Interval Estimate of Large-Sample
Case
• Interval Estimate of Small-Sample
Case
12
Sampling Distribution of
x1  x2
• Properties of the Sampling Distribution of x1  x2
– Expected Value
E ( x1  x2 )  1   2
– Standard Deviation
 x1  x2 
12
n1

 22
n2
where: 1 = standard deviation of population 1
2 = standard deviation of population 2
n1 = sample size from population 1
n2 = sample size from population 2
13
Interval Estimate of 1 - 2:
Large-Sample Case (n1 > 30 and n2 > 30)
• Interval Estimate with 1 and 2 Known
where:
x1  x2  z / 2  x1  x2
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
14
Contoh Soal: Par, Inc.
• 95% Confidence Interval Estimate of the Difference
Between Two Population Means: Large-Sample
Case, 1 and 2 Unknown
Substituting the sample standard deviations for
the population standard deviation:
x1  x2  z / 2
12
 22
(15) 2 ( 20) 2

 17  1. 96

n1 n2
120
80
= 17 + 5.14 or 11.86 yards to 22.14 yards.
We are 95% confident that the difference between the
mean driving distances of Par, Inc. balls and Rap, Ltd.
balls lies in the interval of 11.86 to 22.14 yards.
15
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
1 1
  (  )
n1 n2
2
16
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 s2 (
1 1
1 1
 )  2. 5  2.101 5. 28(  )
n1 n2
12 8
= 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).
17
Inferences About the Difference
Between the Proportions of Two
Populations
• Sampling Distribution of p1  p2
• Interval Estimation of p1 - p2
• Hypothesis Tests about p1 - p2
18
Sampling Distribution of
• Expected Value
p1  p2
E ( p1  p2 )  p1  p2
• Standard Deviation
 p1  p2 
p1 (1  p1 ) p2 (1  p2 )

n1
n2
• Distribution Form
If the sample sizes are large (n1p1, n1(1 - p1),
n2p2,
and n2(1 - p2) are all greater than or equal to 5), the
sampling distribution ofp1  p2 can be approximated
by a normal probability distribution.
19
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 chisquare distribution with n - 1 degrees of
freedom and where 1 -  is the confidence
coefficient.
20
Interval Estimation of 2
• Chi-Square Distribution With Tail Areas of .025
.025
.025
95% of the
possible 2 values
0
2
.975
2
.025
2
21
• Selamat Belajar Semoga Sukses.
22