Normal Distributions 1 1 The Normal Distribution 2 Normal Probability Density Function - < x < Notation: N (, 2 ) A normal distribution with mean and standard deviation 2 2 1 2 1 ) ( - - x e x f The continuous random variable X has a normal distribution if its p.d.f. is 3 Normal Distribution The mean, variance, and m.g.f. of a continuous random variable X that has a normal distribution are: 2 ] [ ] [ X Var X E 4 Normal Distribution 1. “Bell-Shaped” & Symmetrical X f(X) Mean Median Mode 2. Mean, Median, Mode Are Equal 3. Random Variable Has Infinite Range - < x < 5 Example N (72, 5) A normal distribution with mean 72 and variance 5. Possible situations: Test scores, pulse rates, … 6 Effect of Varying Parameters (& ) x f(x) A C B
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Normal Distributions
1
1
The Normal Distribution
2
Normal Probability
Density Function
- < x <
Notation: N (, 2) A normal distribution with
mean and standard deviation
2
2
1
2
1)(
-
-
x
exf
The continuous random variable
X has a normal distribution if
its p.d.f. is
3
Normal Distribution
The mean, variance, and m.g.f. of a
continuous random variable X that has a
normal distribution are:
2][
][
XVar
XE
4
Normal Distribution
1. “Bell-Shaped” &
Symmetrical
X
f(X)
Mean
Median
Mode
2. Mean, Median,
Mode Are Equal
3. Random Variable
Has Infinite Range
- < x <
5
Example
N (72, 5) A normal distribution with mean 72
and variance 5.
Possible situations: Test scores, pulse rates, …
6
Effect of Varying Parameters ( & )
x
f(x)
A C
B
Normal Distributions
2
7
Normal Distribution
Probability
dxxfdxcPd
c )()(
c dx
f(x)
Probability is
area under
curve!
8
X
f(X)
Infinite Number
of Tables
Normal distributions differ by
mean & standard deviation.
Each distribution would
require its own table.
That’s an infinite number!
9
Standard Normal Distribution
Standard Normal Distribution:
A normal distribution with
mean = 0 and standard deviation = 1.
Notation:
Z ~ N ( = 0, 2 = 1)
0 Z
= 1
Cap letter Z 10
Area under Standard Normal Curve
How to find the proportion of the are under the standard normal curve below z or say P ( Z < z ) = ?
Use Standard Normal Table!!!
0 z
Z
11 12
Standard Normal Distribution
P(Z < 0.32) = F(0.32) Area below .32 = ?
0 .32
0.6255
Normal Distributions
3
13
Standard Normal Distribution
P(Z > 0.32) = Area above .32
0 .32
Areas in the upper tail of the standard normal distribution
= 1 - .6255 = .3745
14
Standard Normal Distribution
P(0 < Z < 0.32) = Area between 0 and .32 = ?
0 .32
= 0.6255 – 0.5 = 0.1255
15
-1.00 0 1.00
P ( -1.00 < Z < 1.00 ) = __?___
.3413 .3413
.6826
0.8413 - 0.5 16
-2.00 0 2.00
P ( -2.00 < Z < 2.00 ) = _____
.4772 .4772
.9544 = .4772 + .4772
.9544
17
-3.00 0 3.00
P ( -3.00 < Z < 3.00 ) = _____
… .4987 .4987
.9974
18
- 1.40 0 2.33
P ( -1.40 < Z < 2.33 ) = ____
.4901 .4192
.9093
.0099 .0808
Normal Distributions
4
19
Normal Distribution
= 0
= 1
= 50
= 3
= 70
= 9
…
Standard Normal
20
Standardize the
Normal Distribution
X
Normal
Distribution
ZX
-
One table!
= 0
= 1
Z
Standardized
Normal Distribution
21
Theorem
If X is N(, 2), then Z = is N(0,1). (X – )
-F-
-F
-
-
ab
bZ
aPbXaP )(
22
N ( 0 , 1)
0
-
-
bZ
aPbXaP )(
Standardize the
Normal Distribution
a b
N ( , )
Normal
Distribution
X
Standard
Normal
Distribution
Z
-a
-b
23
Example 1
Normal
Distribution
For a normal distribution that has
a mean = 5 and s.d. = 10, what
percentage of the distribution is
between 5 and 6.2?
X = 5
= 10
6.2 24
Example 1
X= 5
= 10
6.2
Normal
Distribution
12.10
52.6
-
-
XZ
Z= 0
= 1
.12
Standardized
Normal Distribution P(5 X 6.2)
P(0 Z .12)
010
55
-
-
XZ
Normal Distributions
5
25
Z= 0
= 1
.12
Example 1
Standardized Normal
Distribution Table
Area = .5478 - .5 = .0478 26
Example 1
X= 5
= 10
6.2
Normal
Distribution
12.10
52.6
-
-
XZ
Z= 0
= 1
.12
Standardized
Normal Distribution P(5 X 6.2)
P(0 Z .12)
0.0478 4.78%
010
55
-
-
XZ
27
Example
P(3.8 X 5)
X = 5
= 10
3.8
Normal
Distribution
12.10
58.3-
-
-
XZ
Z = 0
= 1
-.12
Standardized
Normal Distribution
.0478
Area = .0478
P(3.8 X 5) =P(-.12 Z 0)
.0478
28
Example
P(2.9 X 7.1)
5
= 10
2.9 7.1 X
Normal
Distribution
0
= 1
-.21 Z.21
Standardized
Normal Distribution
.1664
Area = .0832 + .0832 = .1664
P(2.9 X 7.1) =P(-.21 Z .21)
.1664
29
Example
P(X > 8)
X = 5
= 10
8
Normal
Distribution
Standardized
Normal Distribution
30.10
58
-
-
XZ
Z = 0
= 1
.30
.3821
Area = .3821
P(X > 8) =P(Z > .30)
=.3821
30
X = 5
= 10
8
Normal
Distribution
62% 38%
Value 8 is the 62nd percentile
Example
P(X > 8)
Normal Distributions
6
31
More on Normal Distribution
The work hours per week for residents in Ohio has a
normal distribution with = 42 hours & = 9 hours.
Find the percentage of Ohio residents whose work hours
are
A. between 42 & 60 hours.
P(42 X 60) =?
B. less than 20 hours.
P(X 20) = ?
32
P(42 X 60) = ?
Normal
Distribution
Standardized
Normal Distribution
X
= 200
2400 Z 0
= 1
2.0
9
42 60 2
P(42 Z 60)
P(0 Z 2)
.4772(47.72%) .4772
33
P(X 20) = ?
Normal
Distribution
Standardized
Normal Distribution
X
= 200
2400 Z 0
= 1
2.0
9
20 42 -2.44
P(X 20) = P(Z -2.44)
= 0.0073 = 0.73%
0.0073
34
Finding Z Values
for Known Probabilities
Standardized Normal
Distribution Table
What is z given
P(Z < z) = .80 ?
0
.80 .20
z = .84
z.20 = .84
Def. za : P(Z ≥ za) = a ; P(Z < za) = 1 – a
35
Finding X Values
for Known Probabilities
Example: The weight of new born
infants is normally distributed with a
mean 7 lb and standard deviation of
1.2 lb. Find the 80th percentile.
Area to the left of 80th percentile is 0.800.
In the table, there is a area value 0.7995
(close to 0.800) corresponding to a z-score
of .84.
80th percentile = 7 + .84 x 1.2 = 8.008 lb 36
Finding X Values
for Known Probabilities
Normal Distribution Standardized Normal Distribution
.80 .80
0 7
( )( ) 008.82.184.7 ZX
X = 8.008 z = .84
80th percentile
Normal Distributions
7
37
Stanine Score
1 2 3 4 5 6 7 8 9
-1.75 -1.25 -.75 -.25 0 .25 .75 1.25 1.75
4% 4% ? %
38
More Examples
• The pulse rates for a certain population follow a
normal distribution with a mean of 70 per minute
and s.d. 5. What percent of this distribution that is