Transcript Document

Chapter 3
The Normal Distributions
BPS - 3rd Ed.
Chapter 3
1
Density Curves



Here is a histogram of
vocabulary scores of
947 seventh graders
The smooth curve
drawn over the
histogram is a
mathematical model for
the distribution.
The mathematical
model is called the
density function.
BPS - 3rd Ed.
Chapter 3
2
Density Curves
The areas of the
shaded bars in this
histogram represent the
proportion of scores in
the observed data that
are less than or equal to
6.0.
This proportion is equal
to 0.303.

BPS - 3rd Ed.
Chapter 3
3
Areas under density curves
The
scale of the Y-axis
of the density curve is
adjusted so the total
area under the curve is 1
 The area under the
curve to the left of 6.0 is
shaded, and is equal to
0.293
This similar to the
areas of the shaded bars
in the prior slide!
BPS - 3rd Ed.
Chapter 3
4
Density Curves
BPS - 3rd Ed.
Chapter 3
5
Density Curves
 There
are many types of density curves
 We
are going to focus on a family of
curves called Normal curves
 Normal
curves are
– bell-shaped
– not too steep, not too fat
– defined means & standard deviations
BPS - 3rd Ed.
Chapter 3
6
Normal Density Curves
 The
mean and standard deviation
computed from actual observations
(data) are denoted by x and s,
respectively.
 The
mean and standard deviation of the
distribution represented by the density
curve are denoted by µ (“mu”) and 
(“sigma”), respectively.
BPS - 3rd Ed.
Chapter 3
7
Bell-Shaped Curve:
The Normal Distribution
standard deviation
mean
BPS - 3rd Ed.
Chapter 3
8
The Normal Distribution
Mean µ defines the center of the curve
 Standard deviation  defines the spread
 Notation is N(µ,).

BPS - 3rd Ed.
Chapter 3
9
Practice Drawing Curves!


Symmetrical around μ
Infections points (change in slope, blue arrows) at ± σ
BPS - 3rd Ed.
Chapter 3
10
68-95-99.7 Rule for
Any Normal Curve
 68%
of the observations fall within one
standard deviation of the mean
 95% of the observations fall within two
standard deviations of the mean
 99.7% of the observations fall within
three standard deviations of the mean
BPS - 3rd Ed.
Chapter 3
11
68-95-99.7 Rule for
Any Normal Curve
68%
-
95%
µ +
-2
µ
+2
99.7%
-3
BPS - 3rd Ed.
µ
Chapter 3
+3
12
68-95-99.7 Rule for
Any Normal Curve
BPS - 3rd Ed.
Chapter 3
13
Men’s Height Example (NHANES, 1980)
 Suppose
heights of men follow a Normal
distribution with mean = 70.0 inches and
standard deviation = 2.8 inches
 Shorthand:
X ~ N(70, 2.8)
X  the variable
~  “distributed as”
N(μ, σ)  Normal with mean μ and standard
deviation σ
BPS - 3rd Ed.
Chapter 3
14
Men’s Height Example 68-95-99.7 rule
If X~N(70, 2.8)

68% between µ   = 70.0  2.8 = 67.2 to 72.8

95% between µ  2 = 70.0  2(2.8) = 70.0  5.6 =
64.4 to 75.6 inches

99.7% between µ  3 = 70.0  3(2.8) = 70.0
 8.4 = 61.6 and 78.4 inches
BPS - 3rd Ed.
Chapter 3
15
NHANES (1980) Height Example
What proportion of men are less than 72.8 inches tall?
(Note: 72.8 is one σ above μ on this distribution)
68%
16%
?
-1
+1
84%
BPS - 3rd Ed.
(by 68-95-99.7 Rule)
70
Chapter 3
72.8
(height)
16
NHANES Height Example
What proportion of men are less than 68
inches tall?
?
68 70
(height values)
How many standard deviations is 68 from 70?
BPS - 3rd Ed.
Chapter 3
17
Standard Normal (Z) Distribution
 The
Standard Normal distribution has mean 0
and standard deviation 1
 We call this a Z distribution: Z~N(0,1)
 Any
Normal variable x can be turned into a Z
variable (standardized) by subtracting μ and
dividing by σ:
z
BPS - 3rd Ed.
x

Chapter 3
18
Standardized Scores
 How
many standard deviations is 68
from μ on X~N(70,2.8)?
z
= (x – μ) / σ
= (68  70) / 2.8
= 0.71
 The
value 68 is 0.71 standard
deviations below the mean 70
BPS - 3rd Ed.
Chapter 3
19
Men’s Height Example (NHANES, 1980)
 What
proportion of men are less than
68 inches tall?
?
68 70
-0.71
BPS - 3rd Ed.
0
(height values)
(standardized values)
Chapter 3
20
Table A in text:
Standard Normal Table
BPS - 3rd Ed.
Chapter 3
21
Table A:
Standard Normal Probabilities
z
.00
.01
.02
0.8
.2119
.2090
.2061
0.7
.2420
.2389
.2358
0.6
.2743
.2709
.2676
BPS - 3rd Ed.
Chapter 3
22
Men’s Height Example (NHANES, 1980)
 What
proportion of men are less than
68 inches tall?
.2389
68 70
-0.71
BPS - 3rd Ed.
0
(height values)
(standardized values)
Chapter 3
23
Men’s Height Example (NHANES, 1980)
What proportion of men are greater than 68
inches tall?
 Area under curve sums to 1, so Pr(X > x) = 1
– Pr(X < x), as shown below:

1.2389 =
.2389
.7611
68 70
-0.71
BPS - 3rd Ed.
0
(height values)
(standardized values)
Chapter 3
24
Men’s Height Example (NHANES, 1980)
 How
tall must a man be to place in the
lower 10% for men aged 18 to 24?
.10
? 70
BPS - 3rd Ed.
(height values)
Chapter 3
25
Table A:
Standard Normal Table
 Use
Table A
 Look
up the closest proportion in the table
 Find
corresponding standardized score
 Solve
for X (“un-standardize score”)
BPS - 3rd Ed.
Chapter 3
26
Table A:
Standard Normal Proportion
z
.07
1.3
.0853
1.2
.1020
1.1
.1210
.08
.0838
.1003
.1190
.09
.0823
.0985
.1170
Pr(Z < -1.28) = .1003
BPS - 3rd Ed.
Chapter 3
27
Men’s Height Example (NHANES, 1980)
 How
tall must a man be to place in the
lower 10% for men aged 18 to 24?
.10
? 70
-1.28
BPS - 3rd Ed.
0
(height values)
(standardized values)
Chapter 3
28
Observed Value for a
Standardized Score
 “Unstandardize”
z-score to find
associated x :
z
x
BPS - 3rd Ed.
x    z

Chapter 3
29
Observed Value for a
Standardized Score
x
= μ + zσ
= 70 + (1.28 )(2.8)
= 70 + (3.58)
= 66.42
 A man
would have to be approximately
66.42 inches tall or less to place in the
lower 10% of the population
BPS - 3rd Ed.
Chapter 3
30