Matakuliah
Tahun
: D0722 - Statistika dan Aplikasinya
: 2010
Pendugaan parameter populasi
Pertemuan 6
Learning Outcomes
•
Pada akhir pertemuan ini, diharapkan
mahasiswa akan mampu :
1. menghitung selang kepercayaan bagi
nilai tengah populasi
2. menghitung selang kepercayaan bagi
proporsi, ragam populasi dan ukuran
sample
3
COMPLETE
5th edi tion
1-4
BUSINESS STATISTICS
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
1-5
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
5th edi tion
1-6
BUSINESS STATISTICS
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
5th edi tion
1-7
BUSINESS STATISTICS
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
1-8
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
5th edi tion
1-9
BUSINESS STATISTICS
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
5th edi tion
1-10
BUSINESS STATISTICS
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
5th edi tion
1-11
BUSINESS STATISTICS
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
5th edi tion
1-12
BUSINESS STATISTICS
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
Aczel/Sounderpandian
95% Confidence Interval: n = 40
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
5th edi tion
1-13
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
5th edi tion
1-14
BUSINESS STATISTICS
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
1-15
BUSINESS STATISTICS
5th edi tion
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
BUSINESS STATISTICS
1-16
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
COMPLETE
BUSINESS STATISTICS
1-17
5th edi tion
6-4 Large-Sample Confidence Intervals
for the Population Proportion, p
The estimator of the population proportion, p , is the sample proportion, p . If the
sample size is large, p has an approximately normal distribution, with E( p ) = p and
pq
V( p ) =
, where q = (1 - p). When the population proportion is unknown, use the
n
estimated value, p , to estimate the standard deviation of p .
For estimating p , a sample is considered large enough when both n  p an n  q are greater
than 5.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
5th edi tion
1-18
Large-Sample Confidence Intervals for
the Population Proportion, p
A large - sample (1 -  )100% confidence interval for the population proportion, p :
pˆ  z
α
2
pˆ qˆ
n
where the sample proportion, p̂, is equal to the number of successes in the sample, x,
divided by the number of trials (the sample size), n, and q̂ = 1 - p̂.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
1-19
5th edi tion
Confidence Intervals for the Population
Variance: The Chi-Square (2) Distribution
•
•
The sample variance, s2, is an unbiased estimator
of the population variance, 2.
Confidence intervals for the population variance
are based on the chi-square (2) distribution.
The chi-square distribution is the probability
distribution of the sum of several independent, squared
standard normal random variables.
The mean of the chi-square distribution is equal to the
degrees of freedom parameter, (E[2] = df). The
variance of a chi-square is equal to twice the number of
degrees of freedom, (V[2] = 2df).
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-20
BUSINESS STATISTICS
The Chi-Square (2) Distribution

C hi-S q uare D is trib utio n: d f=1 0 , d f=3 0 , d f =5 0
0 .1 0
df = 10
0 .0 9
0 .0 8
0 .0 7
2

The chi-square random variable cannot
be negative, so it is bound by zero on
the left.
The chi-square distribution is skewed
to the right.
The chi-square distribution approaches
a normal as the degrees of freedom
increase.
f( )

0 .0 6
df = 30
0 .0 5
0 .0 4
df = 50
0 .0 3
0 .0 2
0 .0 1
0 .0 0
0
50
100
2
In sampling from a normal population, the random variable:
 
2
( n  1) s 2
2
has a chi - square distribution with (n - 1) degrees of freedom.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-21
BUSINESS STATISTICS
Values and Probabilities of Chi-Square
Distributions
Area in Right Tail
.995
.990
.975
.950
.900
.100
.050
.025
.010
.005
.900
.950
.975
.990
.995
Area in Left Tail
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
.005
0.0000393
0.0100
0.0717
0.207
0.412
0.676
0.989
1.34
1.73
2.16
2.60
3.07
3.57
4.07
4.60
5.14
5.70
6.26
6.84
7.43
8.03
8.64
9.26
9.89
10.52
11.16
11.81
12.46
13.12
13.79
McGraw-Hill/Irwin
.010
.025
.050
0.000157
0.0201
0.115
0.297
0.554
0.872
1.24
1.65
2.09
2.56
3.05
3.57
4.11
4.66
5.23
5.81
6.41
7.01
7.63
8.26
8.90
9.54
10.20
10.86
11.52
12.20
12.88
13.56
14.26
14.95
0.000982
0.0506
0.216
0.484
0.831
1.24
1.69
2.18
2.70
3.25
3.82
4.40
5.01
5.63
6.26
6.91
7.56
8.23
8.91
9.59
10.28
10.98
11.69
12.40
13.12
13.84
14.57
15.31
16.05
16.79
0.000393
0.103
0.352
0.711
1.15
1.64
2.17
2.73
3.33
3.94
4.57
5.23
5.89
6.57
7.26
7.96
8.67
9.39
10.12
10.85
11.59
12.34
13.09
13.85
14.61
15.38
16.15
16.93
17.71
18.49
.100
0.0158
0.211
0.584
1.06
1.61
2.20
2.83
3.49
4.17
4.87
5.58
6.30
7.04
7.79
8.55
9.31
10.09
10.86
11.65
12.44
13.24
14.04
14.85
15.66
16.47
17.29
18.11
18.94
19.77
20.60
Aczel/Sounderpandian
2.71
4.61
6.25
7.78
9.24
10.64
12.02
13.36
14.68
15.99
17.28
18.55
19.81
21.06
22.31
23.54
24.77
25.99
27.20
28.41
29.62
30.81
32.01
33.20
34.38
35.56
36.74
37.92
39.09
40.26
3.84
5.99
7.81
9.49
11.07
12.59
14.07
15.51
16.92
18.31
19.68
21.03
22.36
23.68
25.00
26.30
27.59
28.87
30.14
31.41
32.67
33.92
35.17
36.42
37.65
38.89
40.11
41.34
42.56
43.77
5.02
7.38
9.35
11.14
12.83
14.45
16.01
17.53
19.02
20.48
21.92
23.34
24.74
26.12
27.49
28.85
30.19
31.53
32.85
34.17
35.48
36.78
38.08
39.36
40.65
41.92
43.19
44.46
45.72
46.98
6.63
9.21
11.34
13.28
15.09
16.81
18.48
20.09
21.67
23.21
24.72
26.22
27.69
29.14
30.58
32.00
33.41
34.81
36.19
37.57
38.93
40.29
41.64
42.98
44.31
45.64
46.96
48.28
49.59
50.89
7.88
10.60
12.84
14.86
16.75
18.55
20.28
21.95
23.59
25.19
26.76
28.30
29.82
31.32
32.80
34.27
35.72
37.16
38.58
40.00
41.40
42.80
44.18
45.56
46.93
48.29
49.65
50.99
52.34
53.67
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
5th edi tion
1-22
Confidence Interval for the Population
Variance
A (1-)100% confidence interval for the population variance * (where the
population is assumed normal) is:

2
2
 ( n  1) s , ( n  1) s 
  2
2  
1


2
2
2

where  is the value of the chi-square distribution with n - 1 degrees of freedom
2
2


that cuts off an area
to its right and
 is the value of the distribution that
cuts off an area of
2
2
1
2
to its left (equivalently, an area of 1 

2
to its right).
* Note: Because the chi-square distribution is skewed, the confidence interval for the
population variance is not symmetric
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
1-23
5th edi tion
6-6 Sample-Size Determination
Before determining the necessary sample size, three questions must
be answered:
• How close do you want your sample estimate to be to the unknown
•
•
parameter? (What is the desired bound, B?)
What do you want the desired confidence level (1-) to be so that the
distance between your estimate and the parameter is less than or equal to
B?
What is your estimate of the variance (or standard deviation) of the
population in question?
n
}
For example: A (1-  ) Confidence Interval for : x  z 

2
Bound, B
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
5th edi tion
1-24
Sample Size and Standard Error
The sample size determines the bound of a statistic, since the standard
error of a statistic shrinks as the sample size increases:
Sample size = 2n
Standard error
of statistic
Sample size = n
Standard error
of statistic

McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
5th edi tion
1-25
Minimum Sample Size:
Mean and Proportion
Minimum required sample size in estimating the population
mean, :
z2 2
n 2 2
B
Bound of estimate:
B = z

2
n
Minimum required sample size in estimating the population
proportion, p
z2 pq
n 2 2
B
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-26
BUSINESS STATISTICS
Sample-Size Determination:
Example
A marketing research firm wants to conduct a survey to estimate the average
amount spent on entertainment by each person visiting a popular resort. The
people who plan the survey would like to determine the average amount spent by
all people visiting the resort to within $120, with 95% confidence. From past
operation of the resort, an estimate of the population standard deviation is
s = $400. What is the minimum required sample size?
z 
2
n
2
2
B
2
2
(196
. ) (400)

120
2
2
 42.684  43
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
RINGKASAN
Penduga parameter nilai tengah :
Selang kepercayaan bagi rata-rata populasi
-ragam diketahui dan tidak diketahui
Selang kepercayaan bagi proporsi populasi
Selang kepercayaan bagi ragam populasi
Penentuan ukuran sampel
27