Download PowerPoint :: 07

CONTINUOUS RANDOM
VARIABLES AND THE NORMAL
DISTRIBUTION
CONTINUOUS PROBABILITY
DISTRIBUTION
Two characteristics
1. The probability that
x assumes a value in any
interval lies in the range 0 to 1
2. The total probability of all the intervals
within which x can assume a value of 1.0
2
Area under a curve between two points.
Shaded area is
between 0 and 1
x=a
x=b
x
3
Total area under a probability distribution curve.
Shaded area is 1.0 or
100%
x
4
Area under the curve as probability.
Shaded area gives the
probability P (a ≤ x ≤ b)
a
b
x
5
The probability of a single value of x is zero.
6
THE NORMAL DISTRIBUTION
Normal Probability Distribution
A normal probability distribution , when
plotted, gives a bell-shaped curve such that
The total area under the curve is 1.0.
2. The curve is symmetric about the mean.
3. The two tails of the curve extend indefinitely.
1.
7
Normal distribution with mean μ and standard deviation σ.
Standard
deviation = σ
Mean = μ
x
8
Total area under a normal curve.
The shaded area is
1.0 or 100%
μ
x
9
A normal curve is symmetric about the mean.
Each of the two shaded
areas is .5 or 50%
.5
.5
μ
x
10
Areas of the normal curve beyond μ ± 3σ.
Each of the two shaded areas is very
close to zero
μ – 3σ
μ
μ + 3σ
x
11
Three normal distribution curves with the
same mean but different standard deviations.
σ=5
σ = 10
σ = 16
μ = 50
x
12
Three normal distribution curves with
different means but the same standard deviation.
σ=5
µ = 20
σ=5
σ=5
µ = 30
µ = 40
x
13
THE STANDARD NORMAL
DISTRIBTUION
Definition
The normal distribution with μ = 0 and σ = 1
is called the standard normal distribution.
14
The standard normal distribution curve.
σ=1
µ=0
-3
-2
-1
0
1
2
3
15
z
THE STANDARD NORMAL
DISTRIBTUION
Definition
The units marked on the horizontal axis of
the standard normal curve are denoted by z
and are called the z values or z scores. A
specific value of z gives the distance between
the mean and the point represented by z in
terms of the standard deviation.
16
Area under the standard normal curve.
Each of these two
areas is .5
.5
-3
-2
-1
.5
0
1
2
3
z
17

Examples Using
The Standard Normal Table …
18
STANDARDIZING A NORMAL
DISTRIBUTION
Converting an x Value to a z Value
For a normal random variable x, a particular value
of x can be converted to its corresponding z value
by using the formula
z
x

where μ and σ are the mean and standard
deviation of the normal distribution of x,
respectively.
19

Examples …
IQ ~ Normal mean = 100 stdev = 15
…

…

20
DETERMINING THE z AND x VALUES WHEN
AN AREA UNDER THE NORMAL
DISTRIBUTION CURVE IS KNOWN
Examples …
21
Finding an x Value for a Normal
Distribution
For a normal curve, with known values of
μ and σ and for a given area under the
curve the x value is calculated as
x = μ + zσ
22