probability continuous 007
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Chapter
Seven
McGraw-Hill/Irwin 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
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Chapter Seven
Continuous Probability Distributions
GOALSWhen you have completed this chapter, you will be able to:ONE
Understand the difference between discrete and continuous
distributions.TWO
Compute the mean and the standard deviation for a uniform
distribution.
THREE
Compute probabilities using the uniform distribution.
FOUR
List the characteristics of the normal probability distribution.
Goals
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Chapter Seven continued
GOALSWhen you have completed this chapter, you will be able to:
FIVE
Define and calculatezvalues.
SIX
Determine the probability an observation will lie between two points
using the standard normal distribution.
SEVEN
Determine the probability an observation will be above or below a
given value using the standard normal distribution.
Continuous Probability Distributions
Goals
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Discrete and continuous
distributions
A Discretedistributionis based on random
variables which canassume only clearly
separated values.
Discrete distributions
studied include:
o Binomial
o Poisson.
A Continuousdistribution usually
results from measuringsomething.
Continuous distributions
include:
o Uniform
o Normalo Others
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The Uniform distribution
Is rectangular in shape
Is defined by minimum and
maximum values
Has a mean computed as
follows:
a + b
2m =
where aand bare
the minimum and
maximum values
Has a standard deviation
computed as follows:
s = (b-a)2
12
The uniform distribution
P(x)
xa b
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Calculates its height as
P(x) = ifa< x < band 0 elsewhere1
(b-a)
Calculates its area as
Area = height* base = *(b-a)1
(b-a)
The uniform distribution
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Suppose the time
that you wait on the
telephone for a liverepresentative of
your phone company
to discuss your
problem with you is
uniformly distributed
between 5 and 25
minutes.
What is the mean wait time?
a+
b
2m==
5+25
2= 15
What is the standarddeviation of the wait time?
s = (b-a)2
12
= (25-5)2
12= 5.77
Example 1
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What is the probability of waiting more than ten minutes?
The area from 10
to 25 minutes is
15 minutes.
Thus:
P(10 < wait time < 25) = height*base
= 1
(25-5) *15 = .75
What is the probability of waiting between 15 and 20
minutes?
The area from 15to 20 minutes is
5 minutes. Thus:
P(15 < wait time < 20) = height*base= 1
(25-5)*5 = .25
Example 2 continued
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is symmetrical about the mean.
is asymptotic. That is the curve gets closer andcloser to the X-axis but never actually touches it.
Has its mean, m, to determine its location andits standard deviation, s, to determine itsdispersion.
The Normal probability distribution
is bell-shaped and has a single peak at the
center of the distribution.
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- 5
0 . 4
0 . 3
0 . 2
0 . 1
. 0
x
f
(
x
r a l i t r b u i o n : m = 0 , s2 = 1
Mean, median, and
mode are equal
Theoretically,
curve extends to
infinity
a
Characteristics of a Normal Distribution
Normal
curve is
symmetrical
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The Standard Normal
Probability Distribution
s
m-
Xz
A z-value is the distance between a selectedvalue, designated X, and the population mean m,divided by the population standard deviation, s.The formula is:
It is also called the
zdistribution.
The standardnormal distribution
is a normal distributionwith a mean of 0 and astandard deviation of 1.
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Example 2
s
m-
X
z
= $2,200 - $2000$200
= 1.00
The bi-monthly
starting salaries of
recent MBAgraduates follows
the normal
distribution with a
mean of $2,000 anda standard deviation
of $200. What is
the z-value for asalary of $2,200?
MBA
3
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EXAMPLE 2 continued
50.1200$
000,2$700,1$-
-
-
s
mXz
What is the
z-value for$1,700?
A z-value of 1 indicates that the value of$2,200 is one standard deviation above the
mean of $2,000. A z-value of1.50 indicatesthat $1,700 is 1.5 standard deviation below themean of $2000.
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Areas Under the Normal
Curve
Practically all is within three standard
deviations of the mean.
m + 3s
About 68 percent of
the area under the
normal curve is withinone standard deviation
of the mean.
m + 1s
About 95 percent is within two standard
deviations of the mean.
m + 2s
7 15
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Example 3
The daily water usage per
person in Providence,
Rhode Island is normallydistributed with a mean of
20 gallons and a standard
deviation of 5 gallons.
About 68 percent of thoseliving in New Providence
will use how many gallons
of water?
About 68% of the daily
water usage will lie between
15 and 25 gallons (+ 1s ).
7 16RHODE
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EXAMPLE 4
00.05
2020
-
-
s
mXz
80.05
2024
-
-
s
mXz
What is the probability that
a person from Providence
selected at random will usebetween 20 and 24 gallons
per day?
Providence
Warwick
Newport
ScituateRes
95
295
RHODE
ISLAND
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Example 4 continued
The area under a normal
curve between az-value of0 and az-value of 0.80 is
0.2881.
We conclude that 28.81
percent of the residents use
between 20 and 24 gallonsof water per day.
See the following diagram
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EXAMPLE 4 continued
40.052018
-
-
-
s
mXz
20.15
2026
-
-
s
mXz
What percent of
the population use
between 18 and 26
gallons per day?
7 20
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EXAMPLE 4 continued
We conclude that 54.03 percent of theresidents use between 18 and 26 gallons
of water per day.
The area
associated with a
z-value of
0.40 is.1554.
The area
associated with a
z-value of 1.20 is.3849.
Adding these
areas, the result is
.5403.
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EXAMPLE 5
Professor Mann has
determined that the scores
in his statistics course areapproximately normally
distributed with a mean of
72 and a standard
deviation of 5. He
announces to the class that
the top 15 percent of the
scores will earn an A.What is the lowest score a
student can earn and still
receive an A?
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EXAMPLE 5 continued
Thez-value associated corresponding to 35
percent is about 1.04.
To begin let X be the score that
separates an A from a B.If 15 percent of the students score
more than X, then 35 percent must
score between the mean of 72 and X.
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EXAMPLE 5 continued
2.772.572)5(04.1725
7204.1
-
X
X
Those with a
score of 77.2 ormore earn anA.
We letzequal 1.04 and
solve the standard normal
equation forX. The resultis the score that separates
students that earned anA
from those that earned aB.
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