Matakuliah
Tahun
: L0104 / Statistika Psikologi
: 2008
Pendugaan Parameter
Nilai Tengah
Pertemuan 13
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa akan dapat menghitung
pendugaan parameter nilai tengah satu
atau dua populasi.
3
Bina Nusantara
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
4
Bina Nusantara
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 Population Proportion

[---------------------
x
x ---------------------]
[--------------------- x ---------------------]
[--------------------- x ---------------------]
Bina Nusantara
Interval Estimate of a Population Mean:
Large-Sample Case (n > 30)
• With σ
Known
x  z /2

n
where: x
is the sample mean
1 -α is the confidence coefficient
zα/2 is the z value providing an area of
α/2 in the upper tail of the standard
normal probability distribution
s
is the population standard deviation
n
is the sample size
Bina Nusantara
Interval Estimate of a Population
Mean:
Large-Sample Case (n > 30)
• With σ 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
Bina Nusantara
n
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 σ is Known
The large-sample interval-estimation procedure can
be used.
• Population is Normally Distributed and σ is Unknown
The appropriate interval estimate is based on a
probability distribution known as the t distribution.
Bina Nusantara
Interval Estimation of a Population
Mean:
Small-Sample Case (n < 30) with σ
Unknown
Interval Estimate
x  t /2
s
n
where 1 -α = the confidence coefficient
tα/2 = the t value providing an area of α/2
in the upper tail of a t distribution
with n - 1 degrees of freedom
s = the sample standard deviation
Bina Nusantara
Interval Estimate of 1 - 2:
Large-Sample Case (n1 > 30 and
n2 > 30)
Interval Estimate with σ1 and σ2 Known
where:
x1  x2  z / 2  x1  x2
1 - α is the confidence coefficient
Interval Estimate with σ1 and σ2 Unknown
x1  x2  z / 2 sx1  x2
where:
sx1  x2
Bina Nusantara
s12 s22


n1 n2
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 –
Bina Nusantara
2
Interval Estimate of μ1 - μ2:
Small-Sample Case (n1 < 30 and/or n2 <
30)
• Interval Estimate with σ 2 Known
where:
x1  x2  z /2 x1  x2
 x1  x2
Bina Nusantara
1 1
  (  )
n1 n2
2
Interval Estimate of μ1 - μ2:
Small-Sample Case (n1 < 30
and/or n2 < 30)
• Interval Estimate with σ 2 Unknown
x1  x2  t /2 sx1  x2
where:
sx1  x2
Bina Nusantara
1
2 1
 s (  )
n1 n2
2
2
(
n

1
)
s

(
n

1
)
s
1
2
2
s2  1
n1  n2  2
Contoh Soal: Specific Motors
• Point Estimate of the Difference Between Two
Population Means
μ1 = mean miles-per-gallon for the population
of M cars
μ2 = mean miles-per-gallon for the population
of J cars
Point estimate of μ1 - μ2 = x1  x2 = 29.8 - 27.3 = 2.5
mpg.
Bina Nusantara
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
1 1
1 1
s (  )  2. 5  2.101 5. 28(  )
n1 n2
12 8
2
= 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).
Bina Nusantara
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.
Bina Nusantara
Contoh Soal: Express Deliveries
Delivery Time (Hours)
District Office
UPX
INTEX
Difference
Seattle
32
25
7
Los Angeles
30
24
6
Boston
19
15
4
Cleveland
16
15
1
New York
15
13
2
Houston
18
15
3
Atlanta
14
15
-1
St. Louis
10
8
2
Milwaukee
7
9
-2
Denver
16
11
5
Bina Nusantara
Contoh Soal: Express Deliveries
• Inference About the Difference Between the Means of
Two Populations: Matched Samples
Let μd = the mean of the difference values for the
two delivery services for the population of
district offices
– Hypotheses
H0: μd = 0, Ha: μd ≠ 0
– Rejection Rule
–
Assuming the population of difference values is
approximately normally distributed, the t distribution with n - 1
degrees of freedom applies. With α = .05, t.025 = 2.262 (9
degrees of freedom).
–
Reject H0 if t < -2.262 or if t > 2.262
Bina Nusantara
Contoh Soal: Express Deliveries
• Inference About the Difference Between the Means of
Two Populations: Matched Samples
 di ( 7  6... 5)
d 

 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
–
Bina Nusantara
Reject H0.
There is a significant difference between the mean delivery
times for the two services.
Selamat Belajar
Semoga Sukses
Bina Nusantara