Matakuliah
Tahun
Versi
: I0284 - Statistika
: 2005
: Revisi
Pertemuan 13
Pendugaan Parameter Nilai Tengah
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa akan dapat menghitung
pendugaan parameter nilai tengah satu
atau dua populasi.
2
Outline Materi
• Penduigaan nilai tengah satu populasi
• Pendugaan beda dua nilai tengah sampel
besar
• Pendugaan beda nilai tengah sampel kecil
• Pendugaan beda nilai tengah populasi
tidak bebas
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 Estimate of a Population Mean:
Large-Sample Case (n > 30)
• With  Known
x  z /2

n
where: 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
5
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
s
deviation.
x  z /2
n
6
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.
7
Interval Estimation of a Population
Mean:
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
8
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
9
Point Estimator of the Difference
Between the Means of Two Populations
Population 1
Par, Inc. Golf Balls
Population 2
Rap, Ltd. Golf Balls
1 = mean driving
2 = mean driving
distance of Rap
golf balls
distance of Par
golf balls
m1 – 2 = difference between
the mean distances
Simple random sample
of n1 Par golf balls
Simple random sample
of n2 Rap golf balls
x1 = sample mean distance
for sample of Par golf ball
x2 = sample mean distance
for sample of Rap golf ball
x1 - x2 = Point Estimate of m1 –
2
10
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
11
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
s 
2
2
( n1  1) s1
2
 ( n2  1) s2
n1  n2  2
12
Contoh Soal: Specific Motors
• Point Estimate of the Difference Between Two
Population Means
1 = mean miles-per-gallon for the population
of M cars
2 = mean miles-per-gallon for the population
of J cars
Point estimate of 1 - 2 = x1  x2 = 29.8 - 27.3
= 2.5 mpg.
13
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).
14
Inference About the Difference
Between the Means of Two
Populations: Matched Samples
• With a matched-sample design each sampled
item provides a pair of data values.
• The matched-sample design can be referred
to as blocking.
• This design often leads to a smaller sampling
error than the independent-sample design
because variation between sampled items is
eliminated as a source of sampling error.
15
Contoh Soal: Express Deliveries
Delivery Time (Hours)
District Office
Seattle
Los Angeles
Boston
Cleveland
New York
Houston
Atlanta
St. Louis
Milwaukee
Denver
UPX
32
30
19
16
15
18
14
10
7
16
INTEX
25
24
15
15
13
15
15
8
9
11
Difference
7
6
4
1
2
3
-1
2
-2
5
16
Contoh Soal: Express Deliveries
• Inference About the Difference Between the Means of
Two Populations: Matched Samples
Let d = the mean of the difference values for the
two delivery services for the population of
district offices
– Hypotheses
H0: d = 0, Ha: d 
– Rejection Rule
Assuming the population of difference values is
approximately normally distributed, the t distribution
with n - 1 degrees of freedom applies. With  = .05,
t.025 = 2.262 (9 degrees of freedom).
Reject H0 if t < -2.262 or if t > 2.262
17
Contoh Soal: Express Deliveries
• Inference About the Difference Between
the Means of Two Populations: Matched
Samples d   di  (7  6... 5)  2. 7
n
10
2
76.1
 ( di  d )
sd 

 2. 9
n 1
9
d  d
2. 7  0
t

 2. 94
sd n 2. 9 10
•
Conclusion
Reject H0.
There is a significant difference between the
mean delivery times for the two services. 18
• Selamat Belajar Semoga Sukses.
19