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 Population
Proportion

[--------------------x ---------------------]
[--------------------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
 n where: is the sample mean
1 x 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 x
 z

/2 s n
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 where 1 -

= the confidence coefficient

/2 t

/2
= the t value providing an area of in the upper tail of a t distribution with n - 1 degrees of freedom s = the sample standard deviation
9

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 -
.025.

= .95,

= .05, and

/2 = 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 of Freedom .10
7
.
.
1.415
8
9
10
.
1.397
1.383
1.372
.
.05
Area in Upper Tail
.025
.01
.
1.895
1.860
1.833
1.812
.
.
2.365
2.306
2.262
2.228
.
.
2.998
2.896
2.821
2.764
.
.005
.
3.499
3.355
3.250
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 x
1
 x
2
• Interval Estimate of 

 


Large-Sample
Case
• Interval Estimate of 

 


Small-Sample
Case
12

Sampling Distribution of
• Properties of the Sampling Distribution of
– Expected Value
E x
1
 x
2
)
 
1
 
2 x
1
 x
2
– Standard Deviation
 x
1
 x
2

 2
1

 n
1 n
2
2
2 where:


1
2
= standard deviation of population 1
= standard deviation of population 2 n
1
= sample size from population 1 n
2
= sample size from population 2 x
1
 x
2
13

Interval Estimate of
Large-Sample Case ( n
1

1

> 30 and
2
: n
2
> 30)
• Interval Estimate with 
1 and

2
Known x
1
  z

/ 2
 x
1
 x
2 where:
1 -
 is the confidence coefficient
• Interval Estimate with 
1 and

2
Unknown x
1
  z

/ 2 s x
1
 x
2 where: s x
1
 x
2 s
1
2
  n
1 s
2
2 n
2 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: x
1
 x
2
 z

/ 2
 2
1
 n
1 n
2
2
 2  
( 15 )
2

( 20 )
120 80
2
= 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
Small-Sample Case ( n
1

1

2
:
< 30 and/or n
2
< 30)
• Interval Estimate with 
2 Known x x z

/ 2


2 where:


2
  2
(
1

1 n
1 n
2
)
16

Contoh Soal: Specific Motors
• 95% Confidence Interval Estimate of the Difference
Between Two Population Means: Small-Sample
Case s
2 
( n
1

1 n
)
1 s
1
2

 n
2
( n

2
2

1 ) s
2
2

11 2 56 )
2 
7 1 81 )
2
 x
1
 x
2
 t
.
025 s
2
(
1

1 n
1 n
2
)
 
.
(
1
12

1
)
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 p
1
 p
2
• Interval Estimation of p
1
p
2
• Hypothesis Tests about p
1
p
2
18

Sampling Distribution of p
2
• Expected Value
E p
1
 p
2
)
 p
1
 p
2
• Standard Deviation
 p
1
 p
2
 p
1
( 1
 n
1 p
1
)
 p
2
( 1
 n
2 p
2
)
• Distribution Form n
2 p
2
,
If the sample sizes are large ( n
1 p
1
, n
1
(1 p
1
), and n
2
(1 p
2
) are all greater than or equal to 5), the sampling distribution of can be approximated by a normal probability distribution.
19

Interval Estimation of  2
• Interval Estimate of a Population Variance
( ( n

1 ) ) s s
/ /
  
( ( n

1 ) ) s s
(
2
( 1
 
/ ) 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
0
.025
95% of the possible  2 values
.025
 2
21

• Selamat Belajar Semoga Sukses.
22