Download Chapter 8 - University of Hawaii at Hilo

Business Statistics
Chapter 8
Bell Shaped Curve
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Describes some data sets
Sometimes called a normal or Gaussian
curve – I’ll use normal
-6
-4
-2
0
2
4
6
8
Central Limit Theorem
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The central limit theorem states that
when an infinite number of successive
random samples are taken from a
population, the distribution of sample
means calculated for each sample will
become approximately normally
distributed
A Family of Curves
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Density Functions
The area under the curve represents
the population
Probabilities can be determined by
viewing the area under the curve
A normal density is specified by its
standard deviation and mean (s and m)
Histogram
45
40
35
Frequency
30
25
20
15
10
5
0
-50
-40
-35
-40
-30
-25
-30
-20
-15
-20
-10
-10-5
0
0
5
Bin
10
10
15
2020
25
30
30
35
40
40
45
More
50
probability density
Normal Curve
.3413
.3413
.1359
.1359
.0215
-3
Approximately
68% of means within +/- 1σ
95% of means within +/- 2σ
99.7% within +/- 3σ
-2
m-2s
-1
m-s
0
m
1
m+s
.0215
2
m+2s
Area under curve = 1.0 (100% of the probability)
3
Some Problems
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A normal distribution has parameters
s = 20 and m = 100
What fraction of values will fall:
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Between 60 and 100?
Between 120 and 140?
Below 50?
What fraction of values will fall:
Between 60 and 100?
0.025
.1359 + .3413 = .4772 = (.4772/1.0000) = 47.72%
0.02
0.015
0.01
.3413
0.005
.1359
0
40
μ-3σ
60
μ-2σ
80
μ-1σ
100
120
μ +1σ
140
μ +2σ
160
μ +3σ
What fraction of values will fall:
Between 120 and 140?
0.025
0.02
0.015
0.01
0.005
.1359
0
40
μ-3σ
60
μ-2σ
80
μ-1σ
100
120
140
μ +1σ
μ +2σ
160
μ +3σ
What fraction of values will fall:
Below 50?
0.025
.5000 or 50%
0.02
0.015
0.01
0.005
.0215
.3413 + .1359 = .4772
0
40
μ-3σ
60
μ-2σ
80
μ-1σ
100
120
μ +1σ
140
μ +2σ
160
μ +3σ
Some Problems (this time with
EXCEL)


A normal distribution has parameters
s = 20 and m = 100
What fraction of values will fall:




Between 60 and 100?
Between 120 and 140?
Below 50?
Above 75?
What fraction of values will fall:
Between 60 and 100?
0.025
NORMDIST(100,100,20,TRUE)-NORMDIST(60,100,20,TRUE)
= .5 -.02275 = .47725
0.02
0.015
0.01
0.005
0
40
60
80
100
120
140
160
What fraction of values will fall:
Between 120 and 140?
0.025
NORMDIST(140,100,20,TRUE)-NORMDIST(120,100,20,TRUE)
0.02
0.015
0.01
0.005
0
40
60
80
100
120
140
160
What fraction of values will fall:
Below 50?
0.025
NORMDIST(50,100,20,TRUE)
0.02
0.015
0.01
0.005
0
40
60
80
100
120
140
160
What fraction of values will fall:
Above 75?
0.025
1 - NORMDIST(75,100,20,TRUE)
0.02
0.015
0.01
0.005
0
40
60
80
100
120
140
160
Some Problems


A normal distribution has parameters
s = 20 and m = 100
What value will:
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50% of the values fall below?
20% of the values fall above?
10% of the values fall above?
What range contains 95% of the values
Normal Curve
probability density
NORMINV(.5,100,20)
50%
-3
-2
-1
0
z-value
1
2
3
Normal Curve
probability density
NORMINV(.2,100,20)
80%
20%
-3
-2
-1
0
z-value
1
2
3
Normal Curve
probability density
NORMINV(.975,100,20)
and
NORMINV(.025,100,20)
95%
2.5%
-3
2.5%
-2
-1
0
z-value
1
2
3
Insurance Sales
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The Great Buffalo Insurance company has
3,000 agents nationwide
Annual sales per agent average $1,500,000
with a standard deviation of $350,000
The sales manager wishes to set a goals such
that 25%, 10%, and 2% of the agents will
exceed the goals
The distribution of sales in normal
Height of Airplane Doors
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Airplane passenger doors are 6 feet in height.
Passenger heights have a normal distribution
with m = 5’6” and s = 6”
What percentage of passengers will need to
duck?
How high should the doors be made so that
only 10% of the passengers must duck?