Download BA 275, Fall 1998 Quantitative Business Methods

BA 275
Quantitative Business Methods
Agenda
 Summarizing Quantitative Data
 The Empirical Rule
 Experiencing Random Behavior


Binomial Experiment
Binomial Probability Distribution
Attention:
Project 1 is due on Wednesday, 1/25/06.
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Quiz #2
Given the data below, complete the following summary statistics table. (Data are in ascending
order): 10.0, 10.5, 12.2, 13.9, 13.9, 14.1, 14.7, 14.7, 15.1, 15.3, 15.9, 17.7, 18.5
2.5
frequency
Count
Average
Median
Variance
Standard deviation
Minimum
Maximum
Range
Lower quartile
Upper quartile
Interquartile range
Sum
1
10
12
14
16
18
20
Variable_X
5
2
1
8
5
3
1.5
0
6
4
2
0.5
6
frequency
frequency
3
Variable X
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14.3462
14.7
5.94936
2.43913
10.0
18.5
8.5
13.9
15.3
1.4
186.5
4
3
2
1
0
8
10
12
14
16
Variable_X
18
20
0
9.99
11.99
13.99
15.99
17.99
Variable_X
19.99
21.99
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Review Example
Given the data below, complete the following summary statistics table. (Data are in ascending
order): 10.0, 10.5, 12.2, 13.9, 13.9, 14.1, 14.7, 14.7, 15.1, 15.3, 15.9, 17.7, 18.5
Count
Average
Median
Variance
Standard deviation
Minimum
Maximum
Range
Lower quartile
Upper quartile
Interquartile range
Sum
Variable X
13
14.3462
14.7
5.94936
2.43913
10.0
18.5
8.5
13.9
15.3
1.4
186.5
Box-and-Whisker Plot
10
12
14
16
18
20
Variable X
Lower invisible line: 11.8
Upper invisible line: 17.4
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The Empirical Rule
99.7%
95%
68%
0.15%
2.35%
13.5%
34%
34%
13.5%
2.35%
0.15%
x  3s
  3
x  2s
xs
x
xs
x  2s
  2
 
 
  2
3
2
1

0
x  3s
  3
1
2
3
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Example
 A set of data whose histogram is bell shaped
yields a sample mean and standard deviation
of 50 and 4, respectively. Approximately what
proportion of observations





Are between 46 and 54?
Are between 42 and 58?
Are between 38 and 62?
Are less than 46?
Are less than 58?
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Example: The Empirical Rule
A manufacturer of automobile batteries
claims that the average length of life for
its grade A battery is 60 months with a
standard deviation of 10 months.
3 cars in your family used this brand of
batteries and none of them lasted for more
than 30 months.
What do you think about the manufacturer’s
claim?
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Example
A manufacturer produces wires with a
mean diameter of 1000 microns, and a
standard deviation of 1 micron. Is a
wire of diameter 1050 microns:
 Fairly likely?
 Pretty unlikely?
 Wildly implausible?
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Example

Suppose that the average hourly earnings of
production workers over the past three years were
reported to be $12.27, $12.85, and $13.39 with the
standard deviations $0.15, $0.18, and $0.23,
respectively. The average hourly earnings of the
production workers in your company also continued
to rise over the past three years from $12.72 in
2002, $13.35 in 2003, to $13.95 in 2004. Assuming
the distribution of the hourly earnings for all
production workers is mound-shaped, demonstrate
quantitatively why the earnings in your company
become less and less competitive.
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Review Example
Year
Industry
average
Industry
std.
2002
12.27
0.15
2003
12.85
0.18
4.73%
13.35
4.95%
2.77
2004
13.39
0.23
4.20%
13.95
4.50%
2.43
%
increase
Company
average
%
increase
12.72
Z score
3
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What statistical lesson can we learn?
“Should we scare the opposition
by announcing our mean height
or lull them by announcing our
median height?”
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Coin-Tossing Example
 n = 10
 p = 0.5
 X = no. of tails in n trials
 Questions




P(X = 5)
P(X = 10)
P(X < 4)
P(X > 8)
 Business Applications?
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Characteristics of a Binomial
Experiment





The experiment consists of n identical trials.
There are only two possible outcomes on each trial.
We will denote one outcome by S (for Success) and
the other by F (for Failure).
The probability of S remains the same from trial to
trial. This probability is denoted by p, and the
probability of F is denoted by 1 – p.
The trials are independent.
The binomial random variable X is the number of
S’s in n trials.
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Is it a Binomial Experiment?



Flip a fair coin 50 times.
Toss a fair die 20 times.
A multiple-choice quiz has 15 questions. Each
question has five possible answers, of which only
one is correct.



Time is running out and you quickly guess all 15
questions without reading them.
You read every question carefully and answer them
to the best of your knowledge.
Among these questions, three of them are main
questions each with 4 subsequent questions. If you
don't know the answer to one question, you won't be
able to answer the subsequent questions correctly.
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Probability Distribution
Binomial Distribution
probability
0.25
Event prob.,Trials
0.5,10
0.2
0.15
0.1
0.05
0
0
2
4
6
8
10
The likelihood of observing
a particular outcome
No. of Successes
All possible outcomes of an experiment
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Binomial Formula and Distribution
 n x
p( X  x)    p (1  p) n  x
 x
Binomial Distribution
probability
0.25
Event prob.,Trials
0.5,10
0.2
0.15
0.1
0.05
0
0
2
4
6
8
10
No. of Successes
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Example
 A sign at a gas station claims that one out of
four cars needs to have oil added. If the
claim is true, what is the probability of the
following events?


One out of the next four cars needs oil.
Two out of the next eight cars need oil.
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