Chapter 6 The Normal Distribution The Normal Distribution The Standardized Normal Distribution

1. © 2003 Prentice-Hall, Inc. Chap 6-1
Basic Business Statistics
(9th
Edition)
Chapter 6
The Normal Distribution

2. © 2003 Prentice-Hall, Inc.
Chap 6-2
Chapter Topics
 The Normal Distribution
 The Standardized Normal Distribution

3. © 2003 Prentice-Hall, Inc.
Chap 6-3
Continuous Probability
Distributions
 Continuous Random Variable
 Values from interval of numbers
 Absence of gaps
 Continuous Probability Distribution
 Distribution of continuous random variable
 Most Important Continuous Probability
Distribution
 The normal distribution

4. © 2003 Prentice-Hall, Inc.
Chap 6-4
Typical Problems this Chapter
will solve
 You have analyzed the earnings per family for 10
communes in Phnom Penh. You know the mean salary
is $60 per family and you have a standard deviation of
10 dollars. When you look at the data in a histogram you
can see is is symmetric (bell shaped – normally
distributed)
In a meeting a sponsor asks you to quickly estimate: -
1. The salaries for the lowest 10% of families and
2. The what percentage earn over $100.

5. © 2003 Prentice-Hall, Inc.
Chap 6-5
Typical Problems this Chapter
will solve
 You have developed a website to sell clothes on-line to US and
European customers. However, very few customers are buying any
products because it is taking so long to down load product pictures.
 Old data indicates you have a mean down load time of 17 seconds
with 3 second standard deviation. Approximately 95% of the
downloads are between 11 and 23 seconds. When you look at the
data in a histogram you can see it is symmetric (bell shaped –
normally distributed)
 The CEO of the clothes company is very angry about your slow
website. You tell him you can use new software that will speed up
the mean down load time by 7 seconds.
If the company buys the new software provide an estimate for: -
1. The time taken for the fastest 10% of downloads and
2. The percentage of downloads that will take over 15 Seconds.

6. © 2003 Prentice-Hall, Inc.
Chap 6-6
The Normal Distribution
 “Bell Shaped”
 Symmetrical
 Mean, Median and
Mode are Equal
 Interquartile Range
Equals 1.33 σ
 Random Variable
Has Infinite Range
Mean
Median
Mode
X
f(X)
µ

7. © 2003 Prentice-Hall, Inc.
Chap 6-7
The Mathematical Model
( ) ( )
( )
( )
2
(1/ 2) /1
2
: density of random variable
3.14159; 2.71828
: population mean
: population standard deviation
: value of random variable
X
f X e
f X X
e
X X
µ σ
πσ
π
µ
σ
− −  
=
≈ ≈
−∞ < < ∞

8. © 2003 Prentice-Hall, Inc.
Chap 6-8
Many Normal Distributions
Varying the Parameters σ and µ, We Obtain
Different Normal Distributions
There are an Infinite Number of Normal Distributions

9. © 2003 Prentice-Hall, Inc.
Chap 6-9
The Standardized Normal
Distribution
When X is normally distributed with a mean and a
standard deviation , follows a
standardized (normalized) normal distribution with a
mean 0 and a standard deviation 1.
X
Z
µ
σ
−
=
X
f(X)
µ
Z
σ
0Zµ =
1Zσ =
f(Z)
µ
σ

10. © 2003 Prentice-Hall, Inc.
Chap 6-10
Z Scores mean the Normal Modal
becomes the Standardized Normal Modal
N(0,1)
Z Scores
µ-3σ µ-2σ µ-1σ µ µ+1σ µ +2σ µ +3σ
•The normal model is
represented by N(µ,σ2
)
Standardized Normal Model
-3 -2 -1 0 1 2 3
•The Standardized Normal
Model is represented by N(0,1)
s
yy
Z
)( −
=
yy
y Z

11. © 2003 Prentice-Hall, Inc.
Chap 6-11
Standardizing
 We often adjust data to find the difference of
each value from a center in terms of a suitable
spread.
 Such standardizing is a fundamental step in
many statistics calculations.

12. © 2003 Prentice-Hall, Inc.
Chap 6-12
Finding Probabilities for Z
Scores
When using the Standard
Normal Distribution the
Probability is the area
under the curve!
c d
X
f(X)
( ) ?P c X d≤ ≤ =
We have generated Z Calc

13. © 2003 Prentice-Hall, Inc.
Chap 6-13
Which Table to Use?
Infinitely Many Normal Distributions
Means Infinitely Many Tables to Look Up!

14. © 2003 Prentice-Hall, Inc.
Chap 6-14
Standardized Normal
Distribution
Z .00 .01
0.0 .5000 .5040 .5080
.5398 .5438
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
.5478
.02
0.1 .5478
Cumulative Standardized
Normal Distribution Table
(Portion)
Probabilities
Only One Table is
0 1Z Zµ σ= =
Z = 0.12
0

15. © 2003 Prentice-Hall, Inc.
Chap 6-15
Finding the Relative Frequency from Tables
 IF Z is positive Use Table E2 Page 811
 If Z is negative Use Table E2 Page 810
Z> Relative Frequency
Z<1.8 0.9641
Z>1.82
Z>1.01
Z Relative Frequency
Z<-0.8 0.2119
Z>-0.82
Z<-1
µ z
Z µ
Table Value
1-(Table Value)
Table Value
1-(Table Value)
Z<
Z>
Z<
Z>

16. © 2003 Prentice-Hall, Inc.
Chap 6-16
Do Examples
 Chapter 6 P221 6.2 , 6.3, 6.4
 Answers for 6.2 and 6.4 are in page 770

17. © 2003 Prentice-Hall, Inc.
Chap 6-17
Standardizing Example
6.2 5
0.12
10
X
Z
µ
σ
− −
= = =
Normal Distribution Standardized
Normal Distribution
10σ = 1Zσ =
5µ =
6.2 X Z
0Zµ =
0.12
calc

18. © 2003 Prentice-Hall, Inc.
Chap 6-18
Example:
Normal Distribution Standardized
Normal Distribution
10σ = 1Zσ =
5µ =
7.1 X Z0Zµ =
0.21
2.9 5 7.1 5
.21 .21
10 10
X X
Z Z
µ µ
σ σ
− − − −
= = = − = = =
2.9 0.21−
.0832
( )2.9 7.1 .1664P X≤ ≤ =
.0832
calc calc

19. © 2003 Prentice-Hall, Inc.
Chap 6-19
Z .00 .01
0.0 .5000 .5040 .5080
.5398 .5438
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
.5832
.02
0.1 .5478
Cumulative Standardized
Normal Distribution Table
(Portion)
0 1Z Zµ σ= =
Z = 0.21
Example:
( )2.9 7.1 .1664P X≤ ≤ = (continued)
0

20. © 2003 Prentice-Hall, Inc.
Chap 6-20
Z .00 .01
-0.3 .3821 .3783 .3745
.4207 .4168
-0.1.4602 .4562 .4522
0.0 .5000 .4960 .4920
.4168.02
-0.2 .4129
Cumulative Standardized
Normal Distribution Table
(Portion)
0 1Z Zµ σ= =
Z = -0.21
Example:
( )2.9 7.1 .1664P X≤ ≤ = (continued)
0

21. © 2003 Prentice-Hall, Inc.
Chap 6-21
Example:
( )8 .3821P X ≥ =
Normal Distribution Standardized
Normal Distribution
10σ = 1Zσ =
5µ =
8 X Z0Zµ =
0.30
8 5
.30
10
X
Z
µ
σ
− −
= = =
.3821
calc

22. © 2003 Prentice-Hall, Inc.
Chap 6-22
Example:
( )8 .3821P X ≥ = (continued)
Z .00 .01
0.0 .5000 .5040 .5080
.5398 .5438
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
.6179.02
0.1 .5478
Cumulative Standardized
Normal Distribution Table
(Portion)
0 1Z Zµ σ= =
Z = 0.30
0

23. © 2003 Prentice-Hall, Inc.
Chap 6-23
Do examples
 6.6,6.8,6.10

24. © 2003 Prentice-Hall, Inc.
Chap 6-24
More Examples of Normal Distribution
A set of final exam grades was found to be normally
distributed with a mean of 73 and a standard deviation of 8.
What is the probability of getting a grade between and 91 on
this exam?
( )2
73,8X N: ( )91 ?P X ≤ =
Mean 73
Standard Deviation 8
X Value 91
Z Value 2.25
P(X<=91) 0.9877756
Probability for X <=
2.250
X
Z
91
8σ =
73µ =
=

25. © 2003 Prentice-Hall, Inc.
Chap 6-25
Example
Mao's height is 1.5 meters (mean heights = 1.4
meters Standard deviation = 2.45meters for males)
What percentage of the male population is higher
(Z>) than Mao? Assume the male heights are
normally distributed
Calculate Z Score (Z calc)
Find Relative Frequency P(X>1.5meters)

26. © 2003 Prentice-Hall, Inc.
Chap 6-26
What percentage of students scored between
65 and 89?
From X Value 65
To X Value 89
Z Value for 65 -1
Z Value for 89 2
P(X<=65) 0.1587
P(X<=89) 0.9772
P(65<=X<=89) 0.8186
Probability for a Range
( )2
73,8X N: ( )65 89 ?P X≤ ≤ =
20
X
Z
8965
-1
73µ =
Examples of Normal Distribution
(continued)
( )2
73,8X N:=

27. © 2003 Prentice-Hall, Inc.
Chap 6-27
2 Types of problems you will have to solve using z
Scores and relative frequencies
Example 1: To find Relative Frequency
In USA SAT Scores are used to grade students. The
mean, µ = 500 and Standard Deviation, σ = 100
N(500,1002
)
 When you are given 2 points in a range and have to
find the proportion of data points (Relative Frequency)
that are between two points.
 What proportion Relative Frequency is between 450
and 600 P(450 =< X<=600)?
1. Draw Normal Model
2. Indicate on the modal the area between 450 and 600
3. Calculate Z calc for 450 and 600
4. Calculate Relative Frequency P(450 =< X<=600)

28. © 2003 Prentice-Hall, Inc.
Chap 6-28
.6217
Finding Z Values for Known
Probabilities
Z .00 0.2
0.0 .5000 .5040 .5080
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
.6179 .6255
.01
0.3
Cumulative Standardized
Normal Distribution Table
(Portion)
What is Z Given
Probability = 0.6217 ?
.6217
0 1Z Zµ σ= =
.31Z =
0

29. © 2003 Prentice-Hall, Inc.
Chap 6-29
Recovering X Values for Known
Probabilities
( ) ( )5 .30 10 8X Zµ σ= + = + =
Normal Distribution
Standardized
Normal Distribution
10σ =
1Zσ =
5µ = ? X Z0Zµ =
0.30
.3821
.6179
This time You Know Mean and
Standard Deviation but not X

30. © 2003 Prentice-Hall, Inc.
Chap 6-30
73µ =
Only 5% of the students taking the test scored
higher than what grade?
( )2
73,8X N: ( )? .05P X≤ =
Cumulative Percentage 95.00%
Z Value 1.644853
X Value 86.15882
Find X and Z Given Cum. Pctage.
1.6450
X
Z
? =86.16
(continued)
More Examples of Normal Distribution
( )2
73,8X N:=

31. © 2003 Prentice-Hall, Inc.
Chap 6-31
Do Examples
 6.8, 6.10, 6.12