Confidence interval

Confidence Intervals

Slide 1

Shakeel Nouman
M.Phil Statistics

Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

6
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Slide 2

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

6-1 Introduction
•

Slide 3

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.

Types of Estimators
•

Slide 4

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.


Confidence Interval or Interval
Estimate

Slide 5

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


6-2 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)



Slide 6

or

0.2

0.1




P x  196
.

n

 
   x  196   0.95
.
n

0.0
-4

-3

-2

-1

0

1

2

3

4

z


6-2 Confidence Interval for 
when
 is Known (Continued)

Slide 7

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% confidenceinterval for  .


A 95% Interval around the
Population Mean

Slide 8

Sampling Distribution of the Mean

Approximately 95% of sample
means can be expected to fall
within the interval
   196  ,   196  
.
.
.

n
n

0.4

95%

f(x)

0.3

0.2

0.1

2.5%



2.5%

0.0

  1.96





  196
.

n



x

n

x

x

2.5% fall below
the interval

x



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

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

 

  1.96
,   1.96
.

n
n





95% Intervals around the
Sample Mean
Sampling Distribution of the Mean

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.)

0.4

95%

f(x)

0.3

0.2

0.1

2.5%

2.5%

0.0

  1.96





  196
.

n

x



x

x

x

*5% of such intervals around the sample

n

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
x

*

Slide 9

x
x
x
x
x
x
x

x

*

x


The 95% Confidence Interval for


Slide 10

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 quantity1.96 
n
the sampling error.

is often called the margin of error or

For example, if:
n = 25 A 95% confidence interval:

20
= 20
x  1.96
 122  1.96
n
25
x = 122

 122  (1.96)(4 )
 122  7.84
 114.16,129.84


A (1- )100% Confidence Interval
for 

Slide 11

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.

æ
ö
P çz > za = a/2
è
ø
2
æ
ö
P çz < - za = a/2
è
ø
2
æ
ö
- za < z < za = (1 - a )
Pç
è 2
ø
2

S t a n d ard N o r m al Dis trib utio n
0.4

(1   )

f(z)

0.3

0.2

0.1





2

2
0.0

-5

-4

-3

-2

-1

z 

2

0

1

2

Z

z
2

3

4

5

(1- a)100% Confidence Interval:
s
x za
n
2


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

z

0.4

(1   )

2

2.576
2.326
1.960
1.645
1.282

0.3

f(z)

(1   )



Slide 12

0.2

0.1





2

2
0.0
-5

-4

-3

-2

-1

z 

2

0

1

2

Z

3

z
2


4

5

The Level of Confidence and the
Width of the Confidence Interval

Slide 13

When sampling from the same population, using a fixed sample
size, the higher the confidence level, the wider the confidence
S t a n d a r d N o r m al Di s tri b u ti o n
interval. S t a n d ar d N o r m al Di s tri b uti o n
0 .3

f(z)

0 .4

0.3

f(z)

0.4

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

80% Confidence Interval:
x  128
.

0

1

2

3

4

Z


n

95% Confidence Interval:
x  196
.



n


5

The Sample Size and the Width
of the Confidence Interval

Slide 14

When sampling from the same population, using a fixed confidence
level, the larger the sample size, n, the narrower the confidence
S
S a m p lin g D is trib u tio n o f th e Me a n
interval.a m p lin g D is trib u tio n o f th e Me a n
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 95% Confidence Interval: n = 40


Example 6-1

Slide 15

• 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.


Example 6-1 (continued) - Using
the Template

Slide 16

Note: The remaining part of the template
display is shown on the next slide.

Example 6-1 (continued) - Using
the Template

Slide 17

 (Sig)

Slide
Example 6-1 (continued) - Using the 18
Template when the Sample Data is
Known


6-3 Confidence Interval or Interval
Estimate for  When  Is Unknown The t Distribution

Slide 19

If the population standard deviation,  is not known, replace
,

with the sample standard deviation, s. If the population is normal,
X 
the resulting statistic:
t 
has a t distribution with (n - 1)
•
•
•

•

s
n
degrees

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

of freedom.
Standard normal
t, df = 20
t, df = 10






The t Distribution Template

Slide 20


6-3 Confidence Intervals for 
when  is Unknown- The t
Distribution

Slide 21

A (1-
)100% confidence interval for when is not
known (assuming a normally distributed population):

x t


2

s
n

where t is the value of the t distribution with n-1 degrees of
2


2
freedom that cuts off a tail area of

to its right.


The t Distribution
t D is trib u tio n : d f= 1 0

-----0 .4

0 .3

Area = 0.10
0 .2

Area = 0.10

}

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
2.358
2.326

}

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
1.980
1.960

f(t)

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
1.658
1.645

0 .1

0 .0
-2.228

Area = 0.025

2.617
2.576

-1.372

0

t

1.372
2.228

}

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
1.289
1.282

}

df t0.100
--13.078
21.886
31.638
41.533
51.476
61.440
71.415
81.397
91.383
101.372
111.363
121.356
131.350
141.345
151.341
161.337
171.333
181.330
191.328
201.325
211.323
221.321
231.319
241.318
251.316
261.315
271.314
281.313
291.311
301.310
401.303
601.296
120

Slide 22

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.



Example 6-2

Slide 23

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 x  10.37% and a standard deviation of s =
3.5%. Assuming a normal population of 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
interval estimate is:
s
x  t 0.025

n
35
.
 10.37  2.145
15
 10.37  1.94
 8.43,12.31


Large Sample Confidence
Intervals for the Population
Mean
df t0.100
--13.078
..
..
..
120



t0.050
----6.314
.
.
.
1.289
1.282

t0.025
----12.706
.
.
.
1.658
1.645

t0.010
-----31.821
.
.
.
1.980
1.960

t0.005
-----63.657
.
.
.
2.358
2.326

------

2.617
2.576

Slide 24

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.


Large Sample Confidence
Mean

Slide 25

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


6-4 Large-Sample Confidence
Proportion, p

Slide 26

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.


6-4 Large-Sample Confidence
Proportion, p

Slide 27

A large - sample (1 -  )100% confidence interval for the population proportion, p :

pz
ˆ

α
2

pq
ˆˆ
n

ˆ
where the sample proportion, p, is equal to the number of successes in the sample, x,
ˆ
ˆ
divided by the number of trials (the sample size), n, and q = 1 - p.


Large-Sample Confidence Interval
for the Population Proportion, p
(Example 6-4)

Slide 28

A marketing research firm wants to estimate the share that foreign companies
have in the American market for certain products. A random sample of 100
consumers is obtained, and it is found that 34 people in the sample are users
of foreign-made products; the rest are users of domestic products. Give a
95% confidence interval for the share of foreign products in this market.

p  z

2

pq
( 0.34 )( 0.66)

 0.34  1.96
n
100
 0.34  (1.96)( 0.04737 )
 0.34  0.0928
  0.2472 ,0.4328

Thus, the firm may be 95% confident that foreign manufacturers control
anywhere from 24.72% to 43.28% of the market.

Large-Sample Confidence Interval for
the Population Proportion, p (Example
6-4) – Using the Template


Reducing the Width of
Confidence Intervals - The Value
of Information

Slide 30

The width of a confidence interval can be reduced only at the price of:
• a lower level of confidence, or
• a larger sample.
Lower Level of Confidence

Larger Sample Size
Sample Size, n = 200

90% Confidence Interval
p  z

2

pq

(0.34)(0.66)
 0.34  1645
.
n
100
 0.34  (1645)(0.04737)
.
 0.34  0.07792
 0.2621,0.4197

p  z

2

pq

(0.34)(0.66)
 0.34  196
.
n
200
 0.34  (196)(0.03350)
.
 0.34  0.0657
 0.2743,0.4057


6-5 Confidence Intervals for the
2
Population Variance: The Chi-Square ( )
Distribution

Slide 31

• The sample variance, s2, is an unbiased
•

2
estimator of the population variance,  .
Confidence intervals for the population
2
variance are based on the chi-square ( )
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
2
the degrees of freedom parameter, (E[ ] = df). The
variance of a chi-square is equal to twice the number
2
of degrees of freedom, (V[ ] = 2df).

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
0 .0 6
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( )



Slide 32

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.

Values and Probabilities of ChiSquare Distributions

Slide 33

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

.010

.025

.050

.100

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

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

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


Values and Probabilities of ChiSquare Distributions

Slide 34


Confidence Interval for the
Population Variance

Slide 35

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
  is the value of the chi-square distribution with n - 1 degrees of freedom
where
2

that cuts off an area  to its right and 2


is the value of the distribution that


1
cuts off an area of  2to its left (equivalently, an area of
2
 to its right).
1
2
2

* Note: Because the chi-square distribution is skewed, the confidence interval for the
population variance is not symmetric

Confidence Interval for the
Population Variance - Example
6-5

Slide 36

In an automated process, a machine fills cans of coffee. If the average amount
filled is different from what it should be, the machine may be adjusted to
correct the mean. If the variance of the filling process is too high, however, the
machine is out of control and needs to be repaired. Therefore, from time to
time regular checks of the variance of the filling process are made. This is done
by randomly sampling filled cans, measuring their amounts, and computing the
sample variance. A random sample of 30 cans gives an estimate s2 = 18,540.
2
Give a 95% confidence interval for the population variance,  .


2
2
 ( n  1) s , ( n 21) s    ( 30  1)18540 , ( 30  1)18540   11765,33604
2

457
.
16.0
   
 


1


2
2


Example 6-5 (continued)

Slide 37

Area in Right Tail
df
.
.
.
28
29
30

.995
.
.
.
12.46
13.12
13.79

.990
.
.
.
13.56
14.26
14.95

.975

.950

.
.
.
15.31
16.05
16.79

.900

.
.
.
16.93
17.71
18.49

.
.
.
18.94
19.77
20.60

.100

.050

.
.
.
37.92
39.09
40.26

.
.
.
41.34
42.56
43.77

.025
.
.
.
44.46
45.72
46.98

.010
.
.
.
48.28
49.59
50.89

.005
.
.
.
50.99
52.34
53.67

Chi-Square Distribution: df = 29
0.06
0.05

0.95

f(2 )

0.04
0.03
0.02

0.025

0.025

0.01
0.00
0

10

20

 2.975  16.05
0

30

40

2

50

60

70

 2.025  4572
.
0


Example 6-5 Using the Template

Slide 38

Using Data


6-6 Sample-Size Determination

Slide 39

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

Sample Size and Standard Error

Slide 40

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




Minimum Sample Size:
Mean and Proportion

Slide 41

Minimum required sample size in estimating the population
mean, m:
z2 s 2
a
n= 2 2
B
Bound of estimate:
s
B = za
n
2
Minimum required sample size in estimating the population
proportion, p
$
z2 pq
a
n= 2 2
B

Sample-Size Determination:
Example 6-6

Slide 42

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

Sample-Size for Proportion:
Example 6-7

Slide 43

The manufacturers of a sports car want to estimate the proportion of people in a
given income bracket who are interested in the model. The company wants to
know the population proportion, p, to within 0.01 with 99% confidence. Current
company records indicate that the proportion p may be around 0.25. What is the
minimum required sample size for this survey?

n

2
z pq
2

B2

2.5762 (0.25)(0.75)

0102
.
 124.42  125

6-7 The Templates – Optimizing
Population Mean Estimates

Slide 44


7 The Templates – Optimizing Population 45
Proportion Estimates


Name
Religion
Domicile
Contact #
E.Mail
M.Phil (Statistics)

Shakeel Nouman
Christian
Punjab (Lahore)
0332-4462527. 0321-9898767
sn_gcu@yahoo.com
sn_gcu@hotmail.com
GC University, .
(Degree awarded by GC University)

M.Sc (Statistics)
Statitical Officer
(BS-17)
(Economics & Marketing
Division)

GC University, .
(Degree awarded by GC University)

Livestock Production Research Institute
Bahadurnagar (Okara), Livestock & Dairy Development
Department, Govt. of Punjab


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