Download 5-2

Chapter 5
Normal Probability
Distributions
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
1
Chapter 5
Normal Probability Distributions
5-1 Overview
5-2 The Standard Normal Distribution
1
5-3 Applications of Normal Distributions
5-2 and 55-3: Normal Distributions: Finding
Values
5-5 The Central Limit Theorem
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
2
Review
x
(# of correct)
0
1
2
3
4
5
0.5
P(x)
.05
.10
.25
.40
.15
.05
Probability
Distribution
0.4
.40
P(x)
0.3
.25
0.2
.15
0.1
0.0
.05
0
.1
1
2
3
4
.05
5
# of correct answers
Probability Histogram
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
3
Overview
™ Continuous random variable
™ Normal distribution
Curve is bell shaped
and symmetric
Figure 55-1
µ
Score
Formula 55-1
y=
e
1
2
σ
( xσ- µ )
2
2π
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
4
5-2
The Standard Normal
Distribution
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
2
5
Definitions
™ Density Curve
(or probability density function)
the graph of a continuous probability
distribution
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
6
Definitions
™ Density Curve
(or probability density function)
the graph of a continuous probability
distribution
1. The total area under the curve must
equal 1.
2. Every point on the curve must have a
vertical height that is 0 or greater.
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
7
Because the total area under
the density curve is equal to 1,
there is a correspondence
between area and probability.
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
3
8
Definition
™ Uniform Distribution
a probability distribution in which the
continuous random variable values are
spread evenly over the range of
possibilities; the graph results in a
rectangular shape.
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
9
Uniform Distributions
Shaded Area = L • W = 4 • 0.2 = 0.8
P( 1º < x < 4º ) = 0.6
P(x )
P( x > 1º ) = 0.8
P(x )
3
0.2
0.2
x
0
0
1
2
3
4
x
0
5
Temperature (degrees Celsius)
0
1
2
3
4
5
Temperature (degrees Celsius)
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
10
Heights of Adult Men and Women
Women:
µ = 63.6
σ = 2.5
4
Men:
µ = 69.0
σ = 2.8
63.6
Figure 5-4
69.0
Height (inches)
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
11
Normal Distributions
Different Locations
µ = 25
σ=8
µ = 100
σ =8
Same Shape
0
25
50
75
Same
Location
µ = 100
σ=8
µ = 100
σ = 15
100
125
Different
Shapes
150
175
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
12
Definition
Standard Normal Distribution
a normal probability distribution that has a
mean of 0 and a standard deviation of 1,
1, and the
total area under its density curve is equal to 1.
-3
-2
-1
0
1
2
3
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
13
Example:
If thermometers have an average
(mean) reading of 0 degrees and a standard deviation
of 1 degree for freezing water and if one thermometer
is randomly selected, find the probability that it reads
freezing water is less than 1.58 degrees.
µ=0
σ=1
5
P(z
P(z < 1.58) = ?
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
14
Methods for Finding Normal
Distribution Areas
™Table AA-2 (sometimes called zzdistribution or zz-table)
™STATDISK
™Other computer programs
™TITI-83 Plus Calculator
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
15
Table A-2
™ Front left cover of text book
™ Formulas and Tables card
™ Appendix
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
NEGATIVE Z Scores
16
Table AA-2
6
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
17
Table A-2
™ Designed only for standard normal distribution
™ Is on two pages: negative z-scores and
positive z-scores
™ Body of table is a cumulative area from the left
up to a vertical boundary
™ Avoid confusion between zz-scores and areas
™ Z-score hundredths is across the top row
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
18
Table AA-2
Standard Normal Distribution
Negative z-scores: cumulative from left
x
0
z
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
19
Table AA-2
Standard Normal Distribution
Positive zz-scores: cumulative from left
7
X
z
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
20
Table AA-2
Standard Normal Distribution
µ=0
σ=1
z=x-µ
σ
X
z
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
21
Table AA-2
Standard Normal Distribution
µ=0
σ=1
z=x-0
1
X
z
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
22
Table AA-2
Standard Normal Distribution
µ=0
σ=1
8
z=x
X
z
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
23
Table AA-2
Standard Normal Distribution
µ=0
σ=1
z=x
Area =
Probability
X
z
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
24
To find:
z Score
the distance along horizontal scale of the
standard normal distribution; refer to the
leftmost column and top row of Table AA-2
Area
the region under the curve; refer to the
values in the body of Table AA-2
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
25
Example:
If thermometers have an average
(mean) reading of 0 degrees and a standard deviation
of 1 degree for freezing water and if one thermometer
is randomly selected, find the probability that it reads
freezing water is less than 1.58 degrees.
µ=0
σ=1
9
P(z
P(z < 1.58) = ?
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
26
POSITIVE z Scores
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
27
Example:
If thermometers have an average (mean)
reading of 0 degrees and a standard deviation of 1 degree
for freezing water and if one thermometer is randomly
selected, find the probability that it reads freezing water is
less than 1.58 degrees.
µ=0
σ=1
P(z
P(z < 1.58) =
0.9429
The probability that the chosen thermometer will measure freezing
freezing
water less than 1.58 degrees is 0.9429.
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
28
Example:
If thermometers have an average (mean)
reading of 0 degrees and a standard deviation of 1 degree
for freezing water and if one thermometer is randomly
selected, find the probability that it reads freezing water is
less than 1.58 degrees.
µ=0
σ=1
10
P(z
P(z < 1.58) =
0.9429
94.29% of the thermometers will read freezing water less than 1.58
1.58
degrees.
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
29
Example:
If we are using the same thermometers,
and if one thermometer is randomly selected, find the
probability that it reads (at the freezing point of water)
above –1.23 degrees.
P (z
(z > –1.23) = ?
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
30
Example:
If we are using the same thermometers,
and if one thermometer is randomly selected, find the
probability that it reads (at the freezing point of water)
above –1.23 degrees.
P (z
(z > –1.23) = 0.8907
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
31
Example:
If we are using the same thermometers,
and if one thermometer is randomly selected, find the
probability that it reads (at the freezing point of water)
above –1.23 degrees.
P (z
(z > –1.23) = 0.8907
11
The probability that the chosen thermometer with a
reading above -1.23 degrees is 0.8907.
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
32
Example:
If we are using the same thermometers,
and if one thermometer is randomly selected, find the
probability that it reads (at the freezing point of water)
above –1.23 degrees.
P (z
(z > –1.23) = 0.8907
The percentage of thermometers with a reading
above -1.23 degrees is 89.07%.
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
33
Example:
A thermometer is randomly selected.
Find the probability that it reads (at the freezing point of
water) between –2.00 and 1.50 degrees.
P (z
(z < –2.00) = 0.0228
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
34
Example:
A thermometer is randomly selected.
Find the probability that it reads (at the freezing point of
water) between –2.00 and 1.50 degrees.
P (z
(z < –2.00) = 0.0228
P (z
(z < 1.50) = 0.9332
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
12
35
Example:
A thermometer is randomly selected.
Find the probability that it reads (at the freezing point of
water) between –2.00 and 1.50 degrees.
P (z
(z < –2.00) = 0.0228
P (z
(z < 1.50) = 0.9332
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
36
Example:
A thermometer is randomly selected.
Find the probability that it reads (at the freezing point of
water) between –2.00 and 1.50 degrees.
P (z
(z < –2.00) = 0.0228
P (z
(z < 1.50) = 0.9332
P (–
(–2.00 < z < 1.50) =
0.9332 – 0.0228 = 0.9104
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
37
Example:
A thermometer is randomly selected.
Find the probability that it reads (at the freezing point of
water) between –2.00 and 1.50 degrees.
P (z
(z < –2.00) = 0.0228
P (z
(z < 1.50) = 0.9332
P (–
(–2.00 < z < 1.50) =
0.9332 – 0.0228 = 0.9104
13
The probability that the chosen thermometer has a
reading between – 2.00 and 1.50 degrees is 0.9104.
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
38
Example:
A thermometer is randomly selected.
Find the probability that it reads (at the freezing point of
water) between –2.00 and 1.50 degrees.
P (z
(z < –2.00) = 0.0228
P (z
(z < 1.50) = 0.9332
P (–
(–2.00 < z < 1.50) =
0.9332 – 0.0228 = 0.9104
The percentage of thermometers that read
between – 2.00 and 1.50 degrees is 91.04%.
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
39
The Empirical Rule
Standard Normal Distribution: µ = 0 and σ = 1
40
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
The Empirical Rule
Standard Normal Distribution: µ = 0 and σ = 1
99.7% of data are within 3 standard deviations of the mean
95% within
2 standard deviations
68% within
1 standard deviation
34%
14
34%
2.4%
2.4%
0.1%
0.1%
13.5%
x - 3s
x - 2s
13.5%
x - 1s
x
x + 1s
x + 2s
x + 3s
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
41
Notation
P(a < z < b)
denotes the probability that the z score is
between a and b
P(z
P(z > a)
denotes the probability that the z score is
greater than a
P (z
(z < a)
denotes the probability that the z score is
less than a
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
42
Notation
P(a < z < b)
between a and b
P(z
P(z > a)
greater than, at least, more than,
not less than
P (z
(z < a)
less than, at most, no more than,
not greater than
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
43
15
Chapter 5. Section 55-1, 55-2. Triola,
Triola, Essentials of Statistics, Second Edition. Copyright 2004. Pearson/Addison
Pearson/Addison--Wesley
44