Download Example 6-1

CHAPTER 6
6-1:Normal
Distribution
Instructor: Alaa saud
Note: This PowerPoint is only a summary and your main source should be the book.
Outline:
Introduction
6-1 Normal Distribution.
6-2 Application of the Normal Distribution.
6-3 The central Limit Theorem.
Note: This PowerPoint is only a summary and your main source should be the book.
 A random variable is a variable whose values are
determined by chance.
Discrete R.V
have a finite No. of
possible value or an
infinite No. of values
can be counted .
CH.(5)
Continuous R.V
Variables that can
assume all values in the
interval between any two
given values .
CH.(6)
Note: This PowerPoint is only a summary and your main source should be the book.
Introduction
• Many continuous variables, have distributions that are
bell-shaped, and these are called approximately normally
distributed variables
Random sample of 100 women
sample size increased and class width decreased
sample size increased and class width decreased
further
Normal Distribution for the population
(b) Negatively skewed
(c) Positively skewed
Mean'
Median
Mode.
(a) Normal
Normal Distribution
The equation for a normal disribution
y
e  2.718
e
 ( X   ) 2 /( 2

2
)
2
  3.14
  population mean
  population s tan dard deviation
Position
parameters
Shape
parameter
(a) Same means but different
standard deviations
(b) Different means but same
standard deviations
(e) Different means and different
standard deviations
A normal distribution is a continuous ,symmetric,
bell-shaped distribution of a variable.
Properties of Normal Distribution :
1.A normal distribution curve is bell shaped.
2.The mean , median and mode are equaled and located at the center .
3.A normal distribution curve is unimodal.
4.The curve is symmetric about the mean .
5.The curve is continuous.
6.The curve never touches the x axis.
7.The total area under a normal distribution curve is equal to 1 or 100%.
8.Embirical Rule .
.
Empirical Rule: Normal Distribution
  3   2   1

68%
95%
99.7
%
  1   2
  3
Standard Normal Distribution :
The equation for a normal disribution
y 
e
 z2 / 2
2
 0
 1
Empirical Rule: Standard Normal Distribution
Procedure To Finding the area under
Standard Normal Distribution:
1. To the left of any Z value
2.To the right of any Z value
P(Z>a)=1-P(Z<a)
P(Z<a)
3.Between any two Z values
P(a<Z<b)=P(Z<b)-P(Z<a)
Example 6-1:
Find the area to the left of z=1.99
P(Z<1.99)=0.9767
Example 6-2:
Find the area to the right of z=-1.16
P(Z>-1.16)=1-P(Z<-1.16)
=1-0.1230
= 0.8770
Example 6-3:
Find the area between z=1.68 and z=-1.37
P(-1.37<Z<1.68)=P(Z<1.68)-P(Z<-1.37)
=0.9535-0.0853
=0.8682
Example 6-4:
Find probability for each
a.P(0<z<2.32)
b.P(z<1.65)
c.P(z>1.91)
P(0<Z<2.32)=P(Z<2.32)-P(Z<0)
=0.9898-0.5000
=0.4898
P(Z<1.65)=0.9505
P(Z>1.91)=1-P(Z<1.91)
=1-0.9719
=0.0281
Example 6-5:
Find the z value such that the area under
the standard normal distribution curve
between 0 and the z value is 0.2123
0.5000+0.2123=0.7123
Z=0.56
Q(41): find the Z value
ANC. Z=-1.39
Q(46):Find the Z value to the right of the mean so that
(b)69.85% of the area under the distribution curve lies to the left of it
ANC. Z=0.52
Q(4 ):Find the Z value to the left of the mean so that :
(a) 98.87% of the area under the distribution curve lies to the right
of it ?
1-0.9887=0.113
Z=-0.28