Matakuliah
Tahun
Versi
: A0064 / Statistik Ekonomi
: 2005
: 1/1
Pertemuan 13
Selang Kepercayaan-1
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Menjelaskan pengertian selang
kepercayaan dan penerapannya bagi
berbagai kondisi populasi
2
Outline Materi
• Selang Kepercayaan bagi μ ketika σ
diketahui
• Selang Kepercayaan bagi μ ketika σ tidak
diketahui
3
COMPLETE
BUSINESS STATISTICS
6







6-4
5th edi tion
Confidence Intervals
Using Statistics
Confidence Interval for the Population Mean When
the Population Standard Deviation is Known
Confidence Intervals for  When  is Unknown - The
t Distribution
Large-Sample Confidence Intervals for the Population
Proportion p
Confidence Intervals for the Population Variance
Sample Size Determination
Summary and Review of Terms
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
6-5
5th edi tion
6-1 Introduction
•
Consider the following statements:
x = 550
•
A single-valued estimate that conveys little information
about the actual value of the population mean.
We are 99% confident that  is in the interval [449,551]
•
An interval estimate which locates the population mean
within a narrow interval, with a high level of confidence.
We are 90% confident that  is in the interval [400,700]
•
An interval estimate which locates the population mean
within a broader interval, with a lower level of confidence.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
6-6
BUSINESS STATISTICS
5th edi tion
Types of Estimators
• Point Estimate
 A single-valued estimate.
 A single element chosen from a sampling distribution.
 Conveys little information about the actual value of the
population parameter, about the accuracy of the estimate.
• Confidence Interval or Interval Estimate
 An interval or range of values believed to include the
unknown population parameter.
 Associated with the interval is a measure of the confidence
we have that the interval does indeed contain the parameter
of interest.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
6-7
5th edi tion
Confidence Interval or Interval Estimate
A confidence interval or interval estimate is a range or interval of
numbers believed to include an unknown population parameter.
Associated with the interval is a measure of the confidence we have
that the interval does indeed contain the parameter of interest.
•
A confidence interval or interval estimate has two
components:
A range or interval of values
An associated level of confidence
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
6-8
BUSINESS STATISTICS
5th edi tion
6-2 Confidence Interval for 
When  Is Known

•
If the population distribution is normal, the sampling
distribution of the mean is normal.
If the sample is sufficiently large, regardless of the shape of
the population distribution, the sampling distribution is
normal (Central Limit Theorem).
In either case:
Standard Normal Distribution: 95% Interval
0.4

 

P   196
.
 x    196
.
  0.95

n
n
f(z)
0.3
or
0.1

 

P x  196
.
   x  196
.
  0.95

n
n
McGraw-Hill/Irwin
0.2
Aczel/Sounderpandian
0.0
-4
-3
-2
-1
0
1
2
3
4
z
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
6-9
BUSINESS STATISTICS
5th edi tion
6-2 Confidence Interval for  when
 is Known (Continued)
Before sampling, there is a 0.95probability that the interval
  1.96

n
will include the sample mean (and 5% that it will not).
Conversely, after sampling, approximately 95% of such intervals
x  1.96

n
will include the population mean (and 5% of them will not).
That is, x  1.96
McGraw-Hill/Irwin

n
is a 95% confidence interval for  .
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
6-10
BUSINESS STATISTICS
5th edi tion
A 95% Interval around the Population
Mean
Sampling Distribution of the Mean
Approximately 95% of sample means
can be expected to fall within the
interval   1.96  ,   1.96  .
0.4
95%
f(x)
0.3
0.2

0.1
2.5%
2.5%
0.0
  196
.


  196
.
n

x
n
x
x
2.5% fall below
the interval
n 
n
Conversely, about 2.5% can be

expected to be above   1.96 n and
2.5% can be expected to be below

  1.96
.
n
x
x
x
2.5% fall above
the interval
x
x
So 5% can be expected to fall outside

 
the interval   196
.
.
,   196
.

n
n
x
x
95% fall within
the interval
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
6-11
BUSINESS STATISTICS
5th edi tion
95% Intervals around the Sample Mean
Sampling Distribution of the Mean
0.4
95%
f(x)
0.3
0.2
0.1
2.5%
2.5%
0.0
  196
.


  196
.
n

x
n
x
x
x
x
x
x
x
x
x
McGraw-Hill/Irwin
x
x
x
*5% of such intervals around the sample
x
*
Approximately 95% of the intervals
 around the sample mean can be
x  1.96
n
expected to include the actual value of the
population mean, . (When the sample
mean falls within the 95% interval around
the population mean.)
*
mean can be expected not to include the
actual value of the population mean.
(When the sample mean falls outside the
95% interval around the population
mean.)
x
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
6-12
BUSINESS STATISTICS
5th edi tion
The 95% Confidence Interval for 
A 95% confidence interval for  when  is known and sampling is
done from a normal population, or a large sample is used:
x  1.96

n

The quantity 1.96
is often called the margin of error or the
n
sampling error.
For example, if: n = 25
 = 20
x = 122
McGraw-Hill/Irwin
A 95% confidence interval:

20
x  1.96
 122  1.96
n
25
 122  (1.96)(4 )
 122  7.84
 114.16,129.84
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
6-13
BUSINESS STATISTICS
5th edi tion
A (1- )100% Confidence Interval for 
We define z as the z value that cuts off a right-tail area of  under the standard
2
2
normal curve. (1-) is called the confidence coefficient.  is called the error
probability, and (1-)100% is called the confidence level.


P z > z   /2


2

 /2


P z
z  


2






P z z z   (1  )
 2

2
S tand ard Norm al Distrib ution
0.4
(1   )
f(z)
0.3
0.2
0.1


2
2
(1- )100% Confidence Interval:
0.0
-5
-4
-3
-2
-1
z 
2
McGraw-Hill/Irwin
0
1
Z
z
2
3
4
5
2
Aczel/Sounderpandian
x  z
2

n
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
6-14
BUSINESS STATISTICS
5th edi tion
Critical Values of z and Levels of
Confidence
0.99
0.98
0.95
0.90
0.80
McGraw-Hill/Irwin
2
0.005
0.010
0.025
0.050
0.100
Stand ard N o rm al Distrib utio n
z
0.4
(1   )
2
2.576
2.326
1.960
1.645
1.282
0.3
f(z)
(1   )

0.2
0.1


2
2
0.0
Aczel/Sounderpandian
-5
-4
-3
-2
-1
z 
2
0
1
Z
z
2
3
4
5
2
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
6-15
BUSINESS STATISTICS
5th edi tion
The Level of Confidence and the Width
of the Confidence Interval
When sampling from the same population, using a fixed sample size, the
higher the confidence level, the wider the confidence interval.
St an d ar d N or m al Di stri b uti o n
0.4
0.4
0.3
0.3
f(z)
f(z)
St an d ar d N or m al Di s tri b uti o n
0.2
0.1
0.2
0.1
0.0
0.0
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
Z
McGraw-Hill/Irwin
1
2
3
4
5
Z
80% Confidence Interval:
x  128
.
0

95% Confidence Interval:
x  196
.
n
Aczel/Sounderpandian

n
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
6-16
BUSINESS STATISTICS
5th edi tion
The Sample Size and the Width of the
Confidence Interval
When sampling from the same population, using a fixed confidence level,
the larger the sample size, n, the narrower the confidence interval.
S a m p lin g D is trib utio n o f th e M e an
S a m p lin g D is trib utio n o f th e M e an
0 .4
0 .9
0 .8
0 .7
0 .3
f(x)
f(x)
0 .6
0 .2
0 .5
0 .4
0 .3
0 .1
0 .2
0 .1
0 .0
0 .0
x
x
95% Confidence Interval: n = 20
McGraw-Hill/Irwin
95% Confidence Interval: n = 40
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
6-17
5th edi tion
Example 6-1
•
Population consists of the Fortune 500 Companies
(Fortune Web Site), as ranked by Revenues. You
are trying to to find out the average Revenues for
the companies on the list. The population standard
deviation is $15,056.37. A random sample of 30
companies obtains a sample mean of $10,672.87.
Give a 95% and 90% confidence interval for the
average Revenues.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
6-18
5th edi tion
Example 6-1 (continued) - Using the
Template
Note: The remaining part of the template display is
shown on the next slide.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
6-19
5th edi tion
Example 6-1 (continued) - Using the
Template
 (Sigma)
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
6-20
5th edi tion
Example 6-1 (continued) - Using the
Template when the Sample Data is
Known
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
6-21
5th edi tion
6-3 Confidence Interval or Interval Estimate
for  When  Is Unknown - The t Distribution
If the population standard deviation, , is not known, replace
 with the sample standard deviation, s. If the population is
normal, the resulting statistic: t  X  
s
n
has a t distribution with (n - 1) degrees of freedom.
•
•
•
•
The t is a family of bell-shaped and symmetric
distributions, one for each number of degree of
freedom.
The expected value of t is 0.
For df > 2, the variance of t is df/(df-2). This is
greater than 1, but approaches 1 as the number
of degrees of freedom increases. The t is flatter
and has fatter tails than does the standard
normal.
The t distribution approaches a standard normal
as the number of degrees of freedom increases
McGraw-Hill/Irwin
Aczel/Sounderpandian
Standard normal
t, df = 20
t, df = 10


© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
6-22
5th edi tion
The t Distribution Template
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
6-23
BUSINESS STATISTICS
5th edi tion
6-3 Confidence Intervals for  when  is
Unknown- The t Distribution
A (1-)100% confidence interval for  when  is not known
(assuming a normally distributed population):
s
x t
n

2
where t is the value of the t distribution with n-1 degrees of
2

freedom that cuts off a tail area of 2 to its right.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
6-24
BUSINESS STATISTICS
5th edi tion
The t Distribution
McGraw-Hill/Irwin
t0.005
-----63.657
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845
2.831
2.819
2.807
2.797
2.787
2.779
2.771
2.763
2.756
2.750
2.704
2.660
2.617
2.576
t D is trib utio n: d f = 1 0
0 .4
0 .3
Area = 0.10
0 .2
Area = 0.10
}
t0.010
-----31.821
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
2.718
2.681
2.650
2.624
2.602
2.583
2.567
2.552
2.539
2.528
2.518
2.508
2.500
2.492
2.485
2.479
2.473
2.467
2.462
2.457
2.423
2.390
2.358
2.326
}
t0.025
-----12.706
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.080
2.074
2.069
2.064
2.060
2.056
2.052
2.048
2.045
2.042
2.021
2.000
1.980
1.960
f(t)
t0.050
----6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
1.711
1.708
1.706
1.703
1.701
1.699
1.697
1.684
1.671
1.658
1.645
0 .1
0 .0
-2.228
Area = 0.025
-1.372
0
t
1.372
2.228
}

t0.100
----3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325
1.323
1.321
1.319
1.318
1.316
1.315
1.314
1.313
1.311
1.310
1.303
1.296
1.289
1.282
}
df
--1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
60
120
Area = 0.025
Whenever  is not known (and the population is
assumed normal), the correct distribution to use is
the t distribution with n-1 degrees of freedom.
Note, however, that for large degrees of freedom,
the t distribution is approximated well by the Z
distribution.
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
6-25
BUSINESS STATISTICS
5th edi tion
Example 6-2
A stock market analyst wants to estimate the average return on a certain
stock. A random sample of 15 days yields an average (annualized) return of
10.37% deviation of s = 3.5%. Assuming a normal population of
andxastandard
returns, give a 95% confidence interval for the average return on this stock.
df
--1
.
.
.
13
14
15
.
.
.
t0.100
----3.078
.
.
.
1.350
1.345
1.341
.
.
.
t0.050
----6.314
.
.
.
1.771
1.761
1.753
.
.
.
McGraw-Hill/Irwin
t0.025
-----12.706
.
.
.
2.160
2.145
2.131
.
.
.
t0.010
-----31.821
.
.
.
2.650
2.624
2.602
.
.
.
t0.005
-----63.657
.
.
.
3.012
2.977
2.947
.
.
.
The critical value of t for df = (n -1) = (15 -1)
=14 and a right-tail area of 0.025 is:
t 0.025  2.145
The corresponding confidence interval or
interval estimate is:x  t 0.025 s
Aczel/Sounderpandian
n
35
.
 10.37  2.145
15
 10.37  1.94
 8.43,12.31
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
6-26
BUSINESS STATISTICS
5th edi tion
Large Sample Confidence Intervals for
the Population Mean
df
--1
.
.
.
120

t0.100
----3.078
.
.
.
1.289
1.282
t0.050
----6.314
.
.
.
1.658
1.645
McGraw-Hill/Irwin
t0.025
-----12.706
.
.
.
1.980
1.960
t0.010
-----31.821
.
.
.
2.358
2.326
t0.005
-----63.657
.
.
.
2.617
2.576
Whenever  is not known (and the population is
assumed normal), the correct distribution to use is
the t distribution with n-1 degrees of freedom.
Note, however, that for large degrees of freedom,
the t distribution is approximated well by the Z
distribution.
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
6-27
5th edi tion
Large Sample Confidence Intervals for
the Population Mean
A large - sample (1 -  )100% confidence interval for :
s
x  z
n
2
Example 6-3: An economist wants to estimate the average amount in checking accounts at banks in a given region. A
random sample of 100 accounts gives x-bar = $357.60 and s = $140.00. Give a 95% confidence interval for , the
average amount in any checking account at a bank in the given region.
x  z 0.025
s
140.00
 357.60  1.96
 357.60  27.44   33016,385
.
.04
n
100
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
Penutup
• Pembahasan materi dilanjutkan dengan
Materi Pokok 14 (Selang Kepercayaan-2)
28