Download Density Curves

+
Density Curves
2
Example: 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 Normal Distributions
Chapter 3
BPS - 5th Ed.
Density Curves
Chapter 3
3
Chapter 3
Density Curves
BPS - 5th Ed.
4
Example: now the area
under the smooth curve
to the left of 6.0 is
shaded. If the scale is
adjusted so the total
area under the curve is
exactly 1, then this
curve is called a density
curve. The proportion
of the area to the left of
6.0 is now equal to
0.293.
Example: 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.
Chapter 3
1
BPS - 5th Ed.
Chapter 3
BPS - 5th Ed.
Density Curves
5
6
Bell-Shaped Curve:
The Normal Distribution
!!The
mean and standard deviation
computed from actual observations (data)
are denoted by x and s, respectively.
!! The
mean and standard deviation of the
population distribution represented by
!
the density curve are denoted by !
(“mu”) and ! (“sigma”), respectively.
Chapter 3
standard deviation
mean
BPS - 5th Ed.
The Normal Distribution
7
Chapter 3
+
BPS - 5th Ed.
8
68-95-99.7 Rule for
Any Normal Curve
Knowing the mean (!) and standard deviation
(!) allows us to make various conclusions
about Normal distributions. Notation: N(!,!).
68%
-!
95%
! +!
-2!
!
+2!
99.7%
-3!
Chapter 3
BPS - 5th Ed.
Chapter 3
!
+3!
BPS - 5th Ed.
+
9
68-95-99.7 Rule for
Any Normal Curve
+
10
Health and Nutrition
Examination Study of
1976-1980
!!Heights
of adult men, aged 18-24
!!mean: 70.0
!!standard
inches
deviation: 2.8 inches
!!heights
follow a normal distribution, so we
have that heights of men are N(70, 2.8).
Chapter 3
+
BPS - 5th Ed.
Health and Nutrition
Examination Study of
1976-1980
!!68-95-99.7
"!68%
11
Rule for men’s heights
Chapter 3
+
!!
What proportion of men are less than 72.8 inches tall?
68%
are between 67.2 and 72.8 inches
16%
are between 64.4 and 75.6 inches
-1
[ ! ± 2! = 70.0 ± 2(2.8) = 70.0 ± 5.6 ]
"!99.7%
are between 61.6 and 78.4 inches
?
? = 84%
[ ! ± 3! = 70.0 ± 3(2.8) = 70.0 ± 8.4 ]
Chapter 3
12
Health and Nutrition
Examination Study of
1976-1980
[ ! ± ! = 70.0 ± 2.8 ]
"!95%
BPS - 5th Ed.
BPS - 5th Ed.
Chapter 3
70
+1
72.8
(height values)
BPS - 5th Ed.
+
13
Health and Nutrition
Examination Study of
1976-1980
What proportion of men are less than 68 inches tall?
+
14
Standardized Scores
!!
How many standard deviations is 68 from 70?
Let’s draw the Population Distribution
!!
and the Standard Normal Curve underneath….
?
68 70
(height values)
How many standard deviations is 68 from 70?
Chapter 3
+
BPS - 5th Ed.
15
Standardized Scores
!!
How many standard deviations is 68 from 70?
!!
standardized score =
(observed value minus mean) / (std dev)
Chapter 3
+
BPS - 5th Ed.
16
Health and Nutrition
Examination Study of
1976-1980
What proportion of men are less than 68 inches tall?
!!
[ = (68 ! 70) / 2.8 = !0.71 ]
!!
The value 68 is 0.71 standard deviations below the mean 70.
?
68 70
-0.71
Chapter 3
BPS - 5th Ed.
Chapter 3
0
(height values)
(standardized values)
BPS - 5th Ed.
+ Table A:
Standard Normal Probabilities
17
!!
See pages 690-691 in text for Table A.
!!
Look up the closest standardized score (z) in the table.
!!
Find the probability (area) to the left of the standardized score.
+ Table A:
Standard Normal Probabilities
18
(the “Standard Normal Table”)
Chapter 3
BPS - 5th Ed.
Table A:
Standard Normal Probabilities
19
z
.00
.01
.02
-0.8
.2119
.2090
.2061
-0.7
.2420
.2389
.2358
Chapter 3
+
BPS - 5th Ed.
20
Health and Nutrition
Examination Study of
1976-1980
What proportion of men are less than 68 inches tall?
!!
.2389
-0.6
.2743
.2709
.2676
68 70
-0.71
Chapter 3
BPS - 5th Ed.
Chapter 3
0
(height values)
(standardized values)
BPS - 5th Ed.
+
21
Health and Nutrition
Examination Study of
1976-1980
What proportion of men are greater than 68 inches tall?
!!
.2389
!!
aged 18 to 24?
0
.10
(height values)
(standardized values)
Chapter 3
+ Table A:
Standard Normal Probabilities
? 70
BPS - 5th Ed.
23
!!
See pages 690-691 in text for Table A.
!!
Look up the closest probability (to .10 here) in the table.
!!
Find the corresponding standardized score.
!!
The value you seek is that many standard deviations from the mean.
Chapter 3
Health and Nutrition
Examination Study of
1976-1980
How tall must a man be to place in the lower 10% for men
22
1-.2389 = .
7611
68 70
-0.71
+
BPS - 5th Ed.
(height values)
Chapter 3
BPS - 5th Ed.
Table A:
Standard Normal Probabilities
Chapter 3
24
z
.07
.08
.09
-1.3
.0853
.0838
.0823
-1.2
.1020
.1003
.0985
-1.1
.1210
.1190
.1170
BPS - 5th Ed.
+
25
Health and Nutrition
Examination Study of
1976-1980
How tall must a man be to place in the lower 10% for men
!!
!!
aged 18 to 24?
? 70
-1.28
0
BPS - 5th Ed.
27
Observed Value for a Standardized
Score
!!
observed value =
mean plus [(standardized score) ! (std dev)]
= 70 + [(!1.28 ) ! (2.8)]
= 70 + (-3.58) = 66.42
!!
Chapter 3
value =
mean plus [(standardized score) ! (std dev)]
(height values)
(standardized values)
Chapter 3
26
Need to “unstandardize” the z-score to find the observed value
(x) :
!! observed
.10
+
Observed Value for a Standardized
Score
A man would have to be approximately 66.42 inches tall or less
to place in the lower 10% of all men in the population.
BPS - 5th Ed.
Chapter 3
BPS - 5th Ed.