Quantitative Methods in Business - Lecture (3)

Quantitative Methods
Dr. Mohamed Ramadan
Moh_Ramadan@icloud.com

Quantitative Methods
Introduction to
Estimation
Lecture (3)

Lecture (3) Introduction to Estimation
The Idea ... Concept and Terminologies
During the annual planning
meeting, the following
scenarios could happen?
Manager: What is your
Sales Target for the Next
Year?
-1- Scenario (1) ... $1.0 million
-2- Scenario (2) ... $0.9 – $1.1 million
Point Estimate
Interval Estimate

The Idea ... Concept and Terminologies
During the annual planning
meeting, the following
scenarios could happen?
Manager: What is your
Sales Target for the Next
Year?
-1- Scenario (1) ... $1.0 million
-2- Scenario (2) ... $0.9 – $1.1 million
Point Estimate
Interval Estimate
Point Estimator
A point estimator draws about a
by the value of an
using a or
.
Interval Estimator
An interval estimator draws about
a by the value of an
using an .

Select Representative
Random Sample
Inference
Objective: Estimate
Average Population Age
Population Parameter
Objective: Estimate
Average Sample Age
Sample Statistic
The objective of
is to
determine the
of a
on
the basis of a
.

Population
Parameters
Sample
Statistics
(Estimates)
𝜇 =
∑!"#
$
𝑥!
𝑁
= &
%&& '
𝑥𝑃 𝑥
𝜎( = &
%&& '
𝑥( 𝑃 𝑥 − 𝜇(
𝑁: 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑠𝑖𝑧𝑒
̅𝑥 =
∑!"#
)
𝑥!
𝑛
= &
%&& '
𝑥𝑃 𝑥
𝑠( =
∑%&& ' 𝑥 − ̅𝑥 (
𝑛 − 1
𝑛: 𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑖𝑧𝑒
𝜎 = 𝜎( 𝑠 = 𝑠(
Sample Estimators to
Population Parameters
Unbiased Estimator
An of a
is an
estimator whose is
to that .
𝜇 = 𝜇 ̅"
𝜇 = 𝐸 ̅𝑥

Population
Parameters
Sample
Statistics
(Estimates)
𝜇 =
∑!"#
$
𝑥!
𝑁
= &
%&& '
𝑥𝑃 𝑥
𝜎( = &
%&& '
𝑥( 𝑃 𝑥 − 𝜇(
̅𝑥 =
∑!"#
)
𝑥!
𝑛
= &
%&& '
𝑥𝑃 𝑥
𝑠( =
∑%&& ' 𝑥 − ̅𝑥 (
𝑛 − 1
𝜎 = 𝜎( 𝑠 = 𝑠(
Consistency
An is said to
be if the
the and the
as the
̅𝑥 − 𝜇 → 𝑠𝑚𝑎𝑙𝑙𝑒𝑟
𝑎𝑠 𝑛 → 𝐿𝑎𝑟𝑔𝑒

Population
Parameters
Sample
Statistics
(Estimates)
𝜇 =
∑!"#
$
𝑥!
𝑁
= &
%&& '
𝑥𝑃 𝑥
𝜎( = &
%&& '
𝑥( 𝑃 𝑥 − 𝜇(
̅𝑥 =
∑!"#
)
𝑥!
𝑛
= &
%&& '
𝑥𝑃 𝑥
𝑠( =
∑%&& ' 𝑥 − ̅𝑥 (
𝑛 − 1
𝜎 = 𝜎( 𝑠 = 𝑠(
Relative Efficiency
If there are
of a , the one
whose is is said to
have
̅𝑥:, 𝑠:
;
& ̅𝑥;, 𝑠;
;
𝑠:
;
< 𝑠;
;
̅𝑥: Relative Efficient

UCL
𝑃 ̅𝑥 − 𝑍!"#
$
𝜎
𝑛
< 𝜇 < ̅𝑥 + 𝑍!"#
$
𝜎
𝑛
= 1 − 𝛼
Estimating Population Mean
Estimating Population Mean 𝜇, when
Population Variance 𝜎 is known?
𝑃 1.2 < 𝜇 < 1.6 = 1 − 0.05
𝑃 1.2 < 𝜇 < 1.6 = 0.95
Confidence Interval
Confidence
Level
Lower Confidence Limit Upper Confidence Limit
LCL
̅𝑥 ± 𝑍#$%
&
𝜎
𝑛
Confidence Interval at confidence
level 1 − 𝛼
A wide interval provides little
information

Cairo Airport Example
Cairo Airport would like to build a 95%
confidence interval for the expected time of
equipping the overseas flights. Therefore, a
sample of 10 flights were randomly observed.
(A) What is the population mean?
(B) Estimate the sample mean?
(C) If we know that the population standard
deviation is 10 minutes, could you inference the
confidence interval for the population mean?
(D) How do we interpret the results?
Flight #
Time (mins)
𝑥
1 20
2 30
3 19
4 17
5 21
6 33
-A- We don’t know the population mean.
Therefore, we will use the above sample to inference
the population mean.
Flight #
Time (mins)
𝑥
7 31
8 30
9 22
10 24
Sum 247

-B-
̅𝑥 =
∑!"#
)
𝑥!
𝑛
=
247
10
= 24.7
Flight #
Time (mins)
𝑥
1 20
2 30
3 19
4 17
5 21
6 33
Flight #
Time (mins)
𝑥
7 31
8 30
9 22
10 24
Sum 247

-C-
̅𝑥 = 24.7 𝑚𝑖𝑛𝑠
𝜎 = 10 𝑚𝑖𝑛𝑠
̅𝑥 ± 𝑍#*+
(
𝜎
𝑛
= 24.7 ± 𝑍,../
(
10
10
= 24.7 ± 𝑍,.01/ 3.2
Flight #
Time (mins)
𝑥
1 20
2 30
3 19
4 17
5 21
6 33
Flight #
Time (mins)
𝑥
7 31
8 30
9 22
10 24
Sum 247

-C-
̅𝑥 = 24.7 𝑚𝑖𝑛𝑠
𝜎 = 10 𝑚𝑖𝑛𝑠
̅𝑥 ± 𝑍#*+
(
𝜎
𝑛
= 24.7 ± 𝑍,.01/ 3.2
= 24.7 ± 1.96×3.2
= 24.7 ± 6.3
Flight #
Time (mins)
𝑥
1 20
2 30
3 19
4 17
5 21
6 33
Flight #
Time (mins)
𝑥
7 31
8 30
9 22
10 24
Sum 247
0.475
0.06
1.9

-C-
̅𝑥 ± 𝑍#*+
(
𝜎
𝑛
= 24.7 ± 6.3
Flight #
Time (mins)
𝑥
1 20
2 30
3 19
4 17
5 21
6 33
Flight #
Time (mins)
𝑥
7 31
8 30
9 22
10 24
Sum 247
24.7 − 6.3 < 𝜇 < 24.7 + 6.3
18.4 < 𝜇 < 31.0

-D- The interpretation is ... We estimate that the
flights will be equipped in average time between 18.4
minutes and 31.0 minutes, and this type of estimator is
correct 95% of the time. That also means that 5% of
the time the estimator will be incorrect.
Flight #
Time (mins)
𝑥
1 20
2 30
3 19
4 17
5 21
6 33
Flight #
Time (mins)
𝑥
7 31
8 30
9 22
10 24
Sum 247
18.4 < 𝜇 < 31.0
95%
2.5%2.5%
𝜇 𝜇

Selecting the Sample Size
𝑃 ̅𝑥 − 𝑍!"#
$
𝜎
𝑛
< 𝜇 < ̅𝑥 + 𝑍!"#
$
𝜎
𝑛
= 1 − 𝛼
Determining the Sample Size?
̅𝑥 − 𝑍!"#
$
𝜎
𝑛
< 𝜇 < ̅𝑥 + 𝑍!"#
$
𝜎
𝑛
̅𝑥 + − ̅𝑥 − 𝑍!"#
$
𝜎
𝑛
< 𝜇 + − ̅𝑥 < ̅𝑥 + − ̅𝑥 + 𝑍!"#
$
𝜎
𝑛
−𝑍!"#
$
𝜎
𝑛
< 𝜇 − ̅𝑥 < 𝑍!"#
$
𝜎
𝑛
𝜇 − ̅𝑥 < 𝑍!"#
$
𝜎
𝑛
limit on the error
of estimation
𝐵 = 𝑍!"#
$
𝜎
𝑛
𝑛 = 𝑍!"#
$
𝜎
𝐵
$

Cohort Sample
A statistics professor wants to compare today’s
students with those 25 years ago. All his current
students’ marks are stored on a computer so that he
can easily determine the population mean. However,
the marks 25 years ago reside only in his musty files.
He does not want to retrieve all the marks and will
be satisfied with a 95% confidence interval
estimate of the mean mark 25 years ago. If he
assumes that the population standard deviation is
12, how large a sample should he take to
estimate the mean to within 2 marks?
Selecting the Sample Size
𝜎 = 12
𝐵 = 2
𝑍,../
(
= 𝑍,.01/ = 1.96
𝑛 = 𝑍#*+
(
𝜎
𝐵
(
= 1.96
12
2
(
= 139 𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠

Quantitative Methods in Business - Lecture (3)

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