 ```6-1
COMPLETE
STATISTICS
by
AMIR D. ACZEL
&amp;
JAYAVEL SOUNDERPANDIAN
6th edition.
6-2
Pertemuan 13 dan 14
Confidence Intervals
6-3
6 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
 The Templates

6-4
6 LEARNING OBJECTIVES
After studying this chapter you should be able to:







Explain confidence intervals
Compute confidence intervals for population means
Compute confidence intervals for population proportions
Compute confidence intervals for population variances
Compute minimum sample sizes needed for an estimation
Compute confidence intervals for special types of
sampling methods
Use templates for all confidence interval and sample size
computations
6-5
6-1 Using Statistics
•
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.
6-6
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.
6-7
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
6-8
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
0.3
f(z)

or

 

P x  196
.
   x  196
.
  0.95

n
n
0.2
0.1
0.0
-4
-3
-2
-1
0
z
1
2
3
4
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

n
is a 95% confidence interval for  .
6-9
6-10
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

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
x
x
95% fall within
the interval
So 5% can be expected to fall outside

 
the interval   196
.
.
,   196
.

n
n
6-11
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
x
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.)
6-12
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
The quantity 1.96
sampling error.

n

n
is often called the margin of error or the
For example, if: n = 25
 = 20
x = 122
A 95% confidence interval:

20
x  1.96
 122  1.96
n
25
 122  (1.96)(4 )
 122  7.84
 114.16,129.84
6-13
A (1-a )100% Confidence Interval for 
We define za as the z value that cuts off a right-tail area of a under the standard
2
2
normal curve. (1-a) is called the confidence coefficient. a is called the error
probability, and (1-a)100% is called the confidence level.


P z &gt; za   a/2


2


P z  za   a/2


2





P  za z za   (1  a)
 2

2
S tand ard Norm al Distrib ution
0.4
(1  a )
f(z)
0.3
0.2
0.1
a
a
2
2
(1- a)100% Confidence Interval:
0.0
-5
-4
-3
-2
-1
z a
2
0
1
Z
za
2
2
3
4
5
x  za
2

n
6-14
Critical Values of z and Levels of
Confidence
0.99
0.98
0.95
0.90
0.80
2
0.005
0.010
0.025
0.050
0.100
Stand ard N o rm al Distrib utio n
za
0.4
(1  a )
2
2.576
2.326
1.960
1.645
1.282
0.3
f(z)
(1  a )
a
0.2
0.1
a
a
2
2
0.0
-5
-4
-3
-2
-1
z a
2
0
1
2
Z
za
2
3
4
5
6-15
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
1
2
3
4
Z
80% Confidence Interval:
x  128
.
0

n
95% Confidence Interval:
x  196
.

n
5
6-16
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
95% Confidence Interval: n = 20
x
95% Confidence Interval: n = 40
6-17
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.
6-18
Example 6-1 (continued) - Using the
Template
Note: The remaining part of the template display is
shown on the next slide.
6-19
Example 6-1 (continued) - Using the
Template
 (Sigma)
6-20
Example 6-1 (continued) - Using the
Template when the Sample Data is Known
6-21
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 &gt; 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
Standard normal
t, df = 20
t, df = 10


6-22
The t Distribution Template
6-23
6-3 Confidence Intervals for  when 
is Unknown- The t Distribution
A (1-a)100% confidence interval for  when  is not known
(assuming a normally distributed population):
s
x t
n
a
2
where ta is the value of the t distribution with n-1 degrees of
2
a
freedom that cuts off a tail area of 2 to its right.
6-24
The t Distribution
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.
6-25
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
.37% deviation of s = 3.5%. Assuming a normal population of
andx a 10
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
.
.
.
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
s
x

t
interval estimate is:
0. 025
n
35
.
 10.37  2.145
15
 10.37  1.94
 8.43,12.31
6-26
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
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.
6-27
Large Sample Confidence Intervals for
the Population Mean
A large - sample (1 - a )100% confidence interval for :
s
x  za
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
```