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Chapter 6
6.1
Normal probability distribution
6.2
6.3
6.4
Standard normal probability distribution
Binomial approximation
Poisson approximation
constructed
from
continuous
random
variables
the entire
area under
the whole
curve is equal
to 1.
Probabilities of
outcome = area
under the curve
between those
points.
What is
Continuous
Probability
Distribution
usually generated
from experiments
in which things are
“measured” as
opposed to
“counted.”
values are
taken on for
every point
over a given
interval
Normal Probability Distribution
• It fits many human characteristics, such as height, weight, length, speed,
IQ, scholastic achievement, and years of life expectancy, among others.
• The normal distribution exhibits the following characteristics.
It is a continuous distribution.
It is a symmetrical distribution about its mean.
It is asymptotic to the horizontal axis.
It is unimodal
It is a family of curves.
Area under the curve is 1.
• The normal distribution is symmetrical.
• Each half of the distribution is a mirror image of the other half.
• Many normal distribution tables contain probability values for only
one side of the distribution because probability values for the other
side of the distribution are identical because of symmetry.
Probability Density Function of the Normal
Distribution
Using Integral Calculus to determine areas under the normal curve from this function is difficult
and time-consuming, therefore, virtually all researchers use table values to analyze normal
distribution problems rather than this formula.
Standardized Normal Distribution
• The conversion formula for any value of a given normal distribution
follows.
• A z score is is the number of standard deviations that a value, x, is above or
below the mean.
• If the value of x <mean, the score is negative;
• if the value of x > mean, the score is positive;
• and if the value of x = mean, the associated score is zero.
How to find the probabilities of Z score?
• The distribution probability values are given in Table A.5.
• By Using calculator Casio fx-570 MS
Press mode 2 times
Press SHIFT
Choose SD (number 1)
Press 3
The area under standard normal distribution defines by P, Q and R
Solving Normal Curve Problems
Example
The Graduate Management Aptitude Test (GMAT), produced by the
Princeton Review in Princeton, New Jersey, is widely used by graduate
schools of business in the United States as an entrance requirement.
Assuming that the scores are normally distributed, probabilities of
achieving scores over various ranges of the GMAT can be determined.
In a recent year, the mean GMAT score was 540 and the standard
deviation was about 100. What is the probability that a randomly
selected score from this administration of the GMAT is between 660
and the mean?
Solution
We must find P(540  x  660)
where   540 and   100
Graphical representation
P(540  x  660)
Standardize the probability into standard normal probability
660  540 
 540  540
P(540  x  660)  P 
Z

100 
 100
 0  Z  1.2
• The probability value in Table 6.2 for z=1.2 is 0.3849.
• Hence the answer is 0.3849
DEMONSTRATION PROBLEM6.7
• Runzheimer International publishes business travel costs for various
cities throughout the world. In particular, they publish per diem
totals, which represent the average costs for the typical business
traveler including three meals a day in business-class restaurants and
single-rate lodging in business-class hotels and motels. If 86.65% of
the per diem costs in Buenos Aires, Argentina, are less than $449 and
if the standard deviation of per diem costs is $36, what is the average
per diem cost in Buenos Aires? Assume that per diem costs are
normally distributed.
Solution
• The standard deviation and an x value are given; the object is to
determine the value of the mean.
• 86.65% = 50 % (the left side) + 36.65 % ( right side)
• So focus on right side, the area = 0.3665. Look the probability in the
table. What is the value of z given the area = 0.3665?
• When z=1.11, the probability is 0.3665
• Hence
Exercise
Exercise
Tompkins Associates reports that the mean clear height for a Class A
warehouse in the United States is 22 feet. Suppose clear heights are
normally distributed and that the standard deviation is 4 feet. A Class A
warehouse in the United States is randomly selected.
a. What is the probability that the clear height is greater than 17 feet?
Answer=0.8944
b. What is the probability that the clear height is less than 13 feet?
Answer=0.0122
c. What is the probability that the clear height is between 25 and 31 feet?
Answer=0.2144
Binomial Approximation
• Some problem in binomial distribution can be approximate by using
normal distribution
• As sample sizes become large, binomial distributions approach the
normal distribution in shape regardless of the value of p.
• This phenomenon occurs faster (for smaller values of n) when p is
near 0.50.
• Let see the simulation here
• Whenever n for any Binomial distribution is
n ≥ 30, and
np>5 and nq>5
Binomial distribution technique is ruled out and will be substituted
with Binomial with Normal Probability distribution.
Continuous Correction Factor (c.c)
• Continuous correction factor need to be made when a continuous
curve is being used to approximate discrete probability distributions.
Continuous Correction Factor
c .c
a) P( X  x) 
 P( x  0.5  X  x  0.5)
c .c
b) P( X  x) 
 P( X  x  0.5)
c .c
c) P( X  x) 
 P( X  x  0.5)
c .c
d) P( X  x) 
 P( X  x  0.5)
c .c
e) P( X  x) 
 P( X  x  0.5)
Example
In a certain country, 45% of registered voters are
male. If 300 registered voters from that country are
selected at random, find the probability that at least
155 are males.
Solutions:
X is the number of male voters.
X ~ b(300, 0.45)
c .c
P( X  155) 
 P( X  155  0.5)  P( X  154.5)
np  300(0.45)  135  5
nq  300(0.55)  165  5

154.5  300(0.45) 
154.5  135 

PZ 
  P  Z 


300(0.45)(0.55)
74.25




 P( Z  2.26)
 0.01191
Exercise (6.23)
According to ars technica, HP is the leading company in the United States in
PC sales with about 27% of the market share. Suppose a business researcher
randomly selects 130 recent purchasers of PCs in the United States.
a.
b.
c.
d.
What is the probability that more than 39 PC purchasers bought an
HP computer?
What is the probability that between 28 and 38 PC purchasers
(inclusive) bought an HP computer?
What is the probability that fewer than 23 PC purchasers bought an
HP computer?
What is the probability that exactly 33 PC purchasers bought an HP
computer?
Poisson Approximation
When the mean  of a Poisson distribution is
relatively large, the normal probability distribution can
be used to approximate Poisson probabilities. A
convenient rule is that such approximation is
acceptable when   10.
Definition
Given a random variable X ~ Po ( ), if   10, then X ~ N ( ,  )
with Z 
X 

Example
A grocery store has an ATM machine inside. An
average of 5 customers per hour comes to use the
machine. What is the probability that more than 30
customers come to use the machine between 8.00
am and 5.00 pm?
Solutions
X is the number of customers come to use the ATM machine in 9 hours.
X ~ Po (45)
  45  10
X ~ N (45, 45)
c .c
P( X  30) 
 P( X  30  0.5)  P( X  30.5)
30.5  45 

PZ 
  P ( Z  2.16)
45 

 0.98461