QBM117
Business Statistics
Probability and Probability Distributions
The Normal Distribution
1
Objectives
•
Learn how to transform a normal random variable to
a standard normal random variable.
•
Calculate probabilities for any normal random
variable.
2
Calculating Probabilities for Any
Normal Distribution
• Probabilities for all normal distributions are calculated
using the standard normal distribution.
• To calculate probabilities for any normal distribution
we transform the normal random variable to a
standard normal random variable and use the
standard normal probability tables (Table 3).
3
Standardizing the Normal Distribution
• A normal random variable X with mean and
standard deviation  is transformed to a standard
normal random variable Z with mean 0 and standard
deviation 1 using the following formula
Z
X 

• Hence if X ~ N (  , 2 ) then the transformation
Z  ( X   ) /  results in Z ~ N (0,1) .
4
Steps for Calculating Probabilities for Any
Normal Distribution
• Sketch the normal curve and mark on the mean 
• Shade the area corresponding to the probability that
you want to find.
• Calculate the z-scores for the boundaries of the
shaded area.
• Sketch the standard normal curve and mark on the
mean at 0.
• Shade the area corresponding to the area on the
normal curve.
• Use table 3 to find the area of the shaded area and
hence the probability that you are looking for.
5
Example 1
Suppose that a random variable X is normally
distributed with a mean of 10 and a standard
deviation of 2.
a. What is the probability that X is less than 9?
b. What is the probability that X is between 10 and
14?
6
2
X ~ N (10, 2 )
X
10
•
What is the probability that X is greater than 12?
P ( X  12 )
X
10 12
8
12  10 

P ( X  12 )  P  Z 

2


 P ( Z  1)
P ( Z  1)
Z
0
1
9
P ( Z  1)  0 . 5  0 . 3413
 0 . 1587
Therefore
P ( X  12 )  P ( X  1)
 0 . 1587
b. What is the probability that X is between 6 and 14?
P (6  X  14)
X
6
10
14
11
14  10 
 6  10
P (6  X  1 4 )  P 
 Z 

2
 2

 P (2  Z  2)
P (  2  Z  2)
2
Z
0
2
12
P (  2  Z  2)  2  P (0  Z  2)
 2  0.4772
 0.9544
T herefore
P (6  Z  14)  P (  2  X  2)
 0.9544
13
Example 2
A consultant was investigating the time it took factory
workers in a car factory to assemble a particular part
after the workers had been trained to perform the
task using an individual learning approach. The
consultant determined that the time in seconds to
assemble the part for workers trained with this
method was normally distributed with a mean of 75
seconds and a standard deviation of 6 seconds.
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a. What is the probability that a randomly selected
factory worker can assemble the part in under 60
seconds?
b. What is the probability that a randomly selected
worker can assemble the part in under 80 seconds?
c. What is the probability that a randomly selected
worker can assemble the part in 65 to 75 seconds?
15
Let X = the time in seconds to assemble the part
2
X ~ N (75, 6 )
X
75
a. What is the probability that a randomly selected
factory worker can assemble the part in under 60
seconds?
P ( X  60 )
60  75 

 PZ 

6


X
60
75
 P ( Z   2.5)
 0.5  0.4938
 0.0062
 2.5
Z
0
17
b. What is the probability that a randomly selected
worker can assemble the part in under 80 seconds?
P ( X  80 )
80  75 

 PZ 

6


 P ( Z  0.83)
75 8 0
X
 0.5  0.2967
 0.7967
0
0.83
Z
18
c. What is the probability that a randomly selected
worker can assemble the part in 65 to 75 seconds?
P (65  X  75)
75  75 
 65  75
 P
 Z 

6
6


65
75
X
0
Z
 P (  1.67  Z  0 )
 0.4525
 1.67
19
Exercise 1
The attendance of football games at a certain
stadium is normally distributed with a mean of 25000
and a standard deviation of 3000.
a. What percentage of the time will attendance be
between 24000 and 28000?
b. What is the probability of the attendance
exceeding 30000?
20
Exercise 2
Mensa is the international high-IQ society. To be a
Mensa member, a person must have an IQ of 132 or
higher. If IQ scores are normally distributed with a
mean of 100 and a standard deviation of 15, what
percentage of the population qualifies for
membership in Mensa?
21
Exercise 3
Battery manufacturers compete on the basis of the
amount of time their product lasts in cameras and
toys. A manufacturer of alkaline batteries has
observed that its batteries last for an average of 26
hours when used in a toy racing car. The amount of
time is normally distributed with a standard deviation
of 2.5 hours.
22
a. What is the probability that a battery lasts
between 24 hours and 28 hours?
b. What is the probability that a battery lasts longer
than 24 hours?
c. What is the probability that a battery lasts less
than 20 hours?
23
Exercise 4
The waiting time at a certain bank is normally
distributed with a mean of 3.7 minutes and a
standard deviation pf 1.4 minutes.
a. What is the probability that a customer has to
wait no more than 2 minutes?
b. What is the probability that a customer has to
wait between 4 and 5 minutes?
24
Exercise 5
The amount spent by students on textbooks in a
semester is normally distributed with a mean of $235
and a standard deviation of $15.
a. What is the probability that a student spends
between $220 and $250 in any semester?
b. What percentage of students spend more than
$270 on textbooks in any semester?
c. What percentage of students spend less than
$225 in a semester?
Exercises
• 5.55
• 5.59
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