Download Standardizing a Normal Distribution

DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION
SYSTEMS
Introduction to Business Statistics
QM 120
Chapter 6
Continuous Probability Distribution
2

When a RV x is discrete, we can assign a positive probability to
each value that x can take and get the probability distribution
for x.

The sum of all probabilities of all values of x is 1.

Not all experiments result in RVs that are discrete.

Continuous RV, such as heights, weights, lifetime of a
particular product, experimental laboratory error…etc. can
assume infinitely many values corresponding to points on line
interval.
Continuous Probability Distribution
3
Suppose we have a set of measurements on a continuous RV,
and we want to create a relative frequency histogram to
describe their distribution.

For a small number of measurements, we can use small
number of classes

Continuous Probability Distribution
4
For a larger number of measurements, we must use a larger
number of classes and reduce the class width

Continuous Probability Distribution
5
As the number of measurements become very large, the class
width become very narrow, the relative frequency histogram
appears more and more like a smooth curve.


This smooth curve describes the probability distribution of a
continuous random variable.
Continuous Probability Distribution
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A continuous RV can take on any of an infinite number of
values on the real line

The depth or density of the probability, which varies with x,
maybe described by a mathematical formula f(x), called the
probability density function for the RV x.

The probability distribution f(x); P(a< x <b) is equal to the
shaded area under the curve.

Continuous Probability Distribution
7

Several important properties of continuous probability
distribution parallel their discrete part.

Just as the sum of discrete probabilities is equal to 1, i.e. P(x)
= 1, and the probability that x falls into a certain interval can
be found by summing the probabilities in that interval,
continuous probability distributions have the following two
characteristics:

The area under a continuous probability distribution is equal to 1.

The probability that x will fall into a particular interval- say, from a
to b – is equal to the area under the curve between the two points a
and b
Continuous Probability Distribution
8

There is a very important difference between the continuous
case and the discrete one which is:

P(x = a) = 0

P(a < x < b) = P(a ≤ x ≤ b)

P( x < a) = P( x ≤ a)
The Normal Distribution
9

Among many continuous probability distributions, the normal
distribution is the most important and widely used one.

A large number of real world phenomena are either exactly or
approximately normally distributed.

The normal distribution or Gaussian distribution is given by a
bell-shaped curve.

A continuous RV x that has a normal distribution is said to be a
normal RV with a mean  and a standard deviation  or simply
x~N(,).

Note that not all the bell-shaped curves represent a normal
distribution.
The Normal Distribution
10

The normal probability distribution, when plotted, gives a bell-
shaped curve such that


The total area under the curve is 1.0.

The curve is symmetric around the mean.

The two tails of the curve extended indefinitely.
The mean  and the standard deviation  are the parameters of
the normal distribution.

Given the values of these two parameters, we can find the area
under the normal curve for any interval.
The Normal Distribution
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There is not just one normal distribution curve but rather a
family of normal distribution.

Each different set of values of  and  gives different normal
curve.

The value of  determines the center of the curve on the
horizontal axis and the value of  gives the spread of the
normal curve

The Normal Distribution
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Like all other distributions, the normal probability
distribution can be expressed by a mathematical function

f (x ) 
1
 2
e
1  x  
 
2   
2
in which the probability that x falls between a and b is the
integral of the above function from a to b, i.e.
b
P (a  x  b )  
a
1
 2
e
1  x  
 
2   
2
dx
The Normal Distribution
13
However we will not use the formula to find the probability.
instead, we will use normal probability table.

The Standard Normal Distribution
14

The standard normal distribution is a special case of the
normal distribution where the  is zero and the  is 1.

The RV that possesses the standard normal distribution is
denoted by z and it is called z values or z scores.
=0
=1
-4
-2
0
z
2
4
The Standard Normal Distribution
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
Since  is zero and the  is 1 for the standard normal, a specific
value of z gives the distance between the mean and the point
represented by z in terms of the standard deviation.

The z values to the right side of the mean are positive and
those on the left are negative BUT the area under the curve is
always positive.

For a value of z = 2, we are 2 standard deviations from the
mean (to the right).

Similarly, for z = -2, we are 2 standard deviations from the
mean (to the left)
The Standard Normal Distribution
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The Standard Normal Distribution
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
The standard normal table lists the area under the standard
normal curve between z = 0 to the values of z to 3.09.

To read the standard normal table, we always start at z = 0,
which represents the mean.

Always remember that the normal curve is symmetric, that is
the area from 0 to any positive z value is equal to the area from 0
to that value in the negative side.
The Standard Normal Distribution
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Example: Find the area under the standard normal curve
between z = 0 to z = 1.95
Example: Find the area under the standard normal curve
between z = -2.17 to z = 0
The Standard Normal Distribution
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Example: Find the following areas under the standard normal
curve.
a) Area to the right of z = 2.32
b) Area to the left of z = -1.54
The Standard Normal Distribution
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Example: Find the following probabilities for the standard
normal curve.
a) P(1.19 < z < 2.12)
b) P(-1.56 < z < 2.31)
c) P(z > -.75)
The Standard Normal Distribution
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
Remember:
When two points are on the same side of the mean, first find
the areas between the mean and each of the two points. Then,
subtract the smaller area from the larger area
When two points are on different side of the mean, first find
the areas between the mean and each of the two points. Then,
add these two area
The Standard Normal Distribution
22

Empirical rule
The Standard Normal Distribution
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Example: Find the following probabilities for the standard
normal curve.
a) P(0 < z < 5.65)
b) P( z < - 5.3)
Standardizing a Normal Distribution
24

As it was shown in the previous section, Table VII of Appendix
C can be used only to find the areas under the standard normal
curve.

However, in real-world applications, most of continuous RVs
that are normally distributed come with mean and standard
deviation different from 0 and 1, respectively.

What shall we do? Is there any way to bring  to zero and  to
1.

Yes, it can be done by subtracting  from x and dividing the
result by  (standardizing)
z 
x 

Standardizing a Normal Distribution
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Example: For the following data [3.2, 1.9, 2.7, 4.3, 3.5, 2.8, 3.9,
4.4, 1.5, 1.8], standardize each value in the sample
x
Transformation
z
3.2
[3.2 - 3]/1
0.2
1.9
[1.9 - 3]/1
-1.1
2.7
[2.7 - 3]/1
-0.3
4.3
[4.3 - 3]/1
1.3
3.5
[3.5 - 3]/1
0.5
2.8
[2.8 - 3]/1
-0.2
3.9
[3.9 - 3]/1
0.9
4.4
[4.4 - 3]/1
1.4
1.5
[1.5 - 3]/1
-1.5
1.8
[1.8 - 3]/1
Mean
3
-1.2
0
St deviation
1
1
Standardizing a Normal Distribution
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Example: Let x be a continuous RV that has a normal
distribution with a mean 50 and a standard deviation of 10.
Convert the following x values to z values
a) x = 55
b) x = 35
c) x = 50
Standardizing a Normal Distribution
27

To find the area between two values of x for a normal
distribution
Convert
both values of x to their respective z values
Find the area under the standard normal curve between those two
values
Example: Let x be a continuous RV that is normally distributed
with a mean of 25 and a standard deviation of 4. Find the area
a) between x = 25 and x = 32
b) between x = 18 and x = 34
Standardizing a Normal Distribution
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Example: Let x be a continuous RV that has a normal
distribution with a mean 40 and a standard deviation of 5. Find
the following probabilities
a) P(x > 55)
b) P(x < 49)
Standardizing a Normal Distribution
29
Example: Let x be a continuous RV that has a normal
distribution with a mean 50 and a standard deviation of 8. Find
the probability P(30 < x < 39)
Standardizing a Normal Distribution
30
Example: Let x be a continuous RV that has a normal
distribution with a mean 80 and a standard deviation of 12. Find
the following probabilities
a) P(70 <x < 135)
b) P(x < 27)
Application of the Normal Distribution
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Example: According to the U.S. Bureau of Census, the mean
income of all U.S. families was $43,237 in 1991. Assume that the
1991 incomes of all U.S. families have a normal distribution with
a mean of $43,237 and a standard deviation of $10,500. Find the
probability that the 1991 income of a randomly selected U.S.
family was between $30,000 and $50,000.
Application of the Normal Distribution
32
Example: A racing car is one of the many toys manufactured by
Mack Corporation. The assembly time for this toy follows a
normal distribution with a mean of 55 minutes and a standard
deviation of 4 minutes. The company closes at 5 P.M. every day.
If one worker starts assembling a racing car at 4 P.M., what is
the probability that she will finish this job before the company
closes for the day?
Application of the Normal Distribution
33
Example: Hupper Corporation produces many types of soft
drinks, including Orange Cola. The filling machines are
adjusted to pour 12 ounces of soda in each 12-ounce can of
Orange Cola. However, the actual amount of soda poured into
each can is not exactly 12 ounces; it varies from can to can. It is
found that the net amount of soda in such a can has a normal
distribution with a mean of 12 ounces and a standard
deviation of .015 ounces.
(a) What is the probability that a randomly selected can of
Orange Cola contains 11.97 to 11.99 ounces of soda?
(b) What percentage of the Orange Cola cans contain 12.02 to
12.07 ounces of soda?
Application of the Normal Distribution
34
Example: The life span of a calculator manufactured by lntal
Corporation has a normal distribution with a mean of 54
months and a standard deviation of 8 months. The company
guarantees that any calculator that starts malfunctioning within
36 months of the purchase will be replaced by a new one. About
what percentage of such calculators made by this company are
expected to be replaced?
Determining the z and x values
35

We reverse the procedure of finding the area under the normal
curve for a specific value of z or x to finding a specific value of z
or x for a known area under the normal curve.
Example: Find a point z such that the area under the standard
normal curve between 0 and z is .4251 and the value of z is
positive
Determining the z and x values
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Determining the z and x values
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Example: Find the value of z such that the area under the
standard normal curve in the right tail is .0050.
Determining the z and x values
38
Example: Find the value of z such that the area under the
standard normal curve in the left tail is .050.
Determining the z and x values
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Finding an x value for a normal distribution:
To find an x value when an area under a normal distribution
curve is given, we do the following
1. Find the z value corresponding to that x value from the
standard normal curve.
2. Transform the z value to x by substituting the values of , ,
and z in the following formula
x    z
Determining the z and x values
40
Example: Recall the calculators example, it is known that the life
of a calculator manufactured by Intal Corporation has a normal
distribution with a mean of 54 months and a standard deviation
of 8 months. What should the warranty period be to replace a
malfunctioning calculator if the company does not want to
replace more than 1 % of all the calculators sold
Determining the z and x values
41
Example: Most business schools require that every applicant for
admission to a graduate degree program take the GMAT.
Suppose the GMAT scores of all students have a normal
distribution with a mean of550 and a standard deviation of 90.
Matt Sanger is planning to take this test soon. What should his
score be on this test so that only 10% of all the examinees score
higher than he does?
Determining the z and x values
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Example: Find the value of z such that 0.95 of the area is within
±z standard deviations of the mean