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Problem 7.
For a bell-shaped distributed trait, determine the
approximate percentage of the
population falling within each of the following z-score
intervals:
(a) (0,.56)
(b) (0,1.57)
(c) (-1.74,0)
(d) (.50,1.67)
(e) (-1.43, 2.1)
(f) -2.01<z<-.50
(g) z>1.52
(h) z>-1.03
(i) z<1.52
(j) z<-1.03
Previous Definitions
30. Approximately
equals is denoted by
 .
31. Percentages
for a normal distribution and of a population:
a. The symbol %a(z1,z2)
will be used to denote the percentage of the area under a normal
curve (bell curve) that falls above the z-score interval (z1,z2).
b. The symbol %p(z1,z2)
will be used to denote the percentage of a population with a
trait falling within the
z-score interval (z1,z2).
32. A population trait
that has a histogram that is bell-shaped is often referred to as a
bell-shaped distributed trait.
Note that by these definitions and
the symmetry of the normal curve, we have, for a population with a
bell-shaped distributed trait, %p(z1,z2)
%a(z1,z2),
%p(0, )
%a(0,)
= 50%, and %p(-,0)
%a(-,0)
= 50%.
Solution
(a) As
the trait is bell-shaped,
we have for the interval (0,.56):
The percentage
of the population falling within (0,.56) = %p(0,.56)
%a(0,.56) = the area depicted within:
We use the
table in
Centrality,
Spreads, Normality, and Chebyshev's Rule
to determine the percentage. We go down to row 0.5
and over to column 6 to obtain the percentage. The applicable part
of the table appears
below with the applicable cell highlighted.
|
Part of
Table
I: Table of Percentages
Corresponding to Z-scores |
|
|
|
z to one digit to the
right of the decimal point |
z's second digit to the
right of the decimal point |
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
0.0 |
0.0% |
0.4% |
0.8% |
1.2% |
1.6% |
2.0% |
2.4% |
2.8% |
3.2% |
3.6% |
|
|
|
|
|
|
|
|
|
|
|
|
|
0.4 |
15.5% |
15.9% |
16.3% |
16.6% |
17.0% |
17.4% |
17.7% |
18.1% |
18.4% |
18.8% |
|
0.5 |
19.1% |
19.5% |
19.8% |
20.2% |
20.5% |
20.9% |
21.2% |
21.6% |
21.9% |
22.2% |
|
0.6 |
22.6% |
22.9% |
23.2% |
23.6% |
23.9% |
24.2% |
24.5% |
24.9% |
25.2% |
25.5% |
|
|
|
|
|
|
|
|
|
|
|
|
|
3.0 |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
|
Note: The unit
of measure of the z-score is standard deviation. |
|
Sources:
Statistics for Business and Economics: Eighth Edition,
p. 987, and Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, pp. 966-972. |
Hence, the
percentage of the population falling within (0,.56)
21.2%.
(b) As
the trait is bell-shaped,
we have for the interval (0,1.57):
The percentage
of the population falling within (0,1.57) = %p(0,1.57)
%a(0,1.57) = the area depicted within:
We use the
table in
Centrality,
Spreads, Normality, and Chebyshev's Rule
to determine the percentage. We go down to row 1.5
and over to column 7 to obtain the percentage. The applicable part
of the table appears
below with the applicable cell highlighted.
|
Part of
Table
I: Table of Percentages
Corresponding to Z-scores |
|
|
|
z to one digit to the
right of the decimal point |
z's second digit to the
right of the decimal point |
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
0.0 |
0.0% |
0.4% |
0.8% |
1.2% |
1.6% |
2.0% |
2.4% |
2.8% |
3.2% |
3.6% |
|
|
|
|
|
|
|
|
|
|
|
|
|
1.4 |
41.9% |
42.1% |
42.2% |
42.4% |
42.5% |
42.6% |
42.8% |
42.9% |
43.1% |
43.2% |
|
1.5 |
43.3% |
43.4% |
43.6% |
43.7% |
43.8% |
43.9% |
44.1% |
44.2% |
44.3% |
44.4% |
|
1.6 |
44.5% |
44.6% |
44.7% |
44.8% |
44.9% |
45.1% |
45.2% |
45.3% |
45.4% |
45.4% |
|
|
|
|
|
|
|
|
|
|
|
|
|
3.0 |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
|
Note: The unit
of measure of the z-score is standard deviation. |
|
Sources:
Statistics for Business and Economics: Eighth Edition,
p. 987, and Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, pp. 966-972. |
Hence, the
percentage of the population falling within (0,1.57)
44.2%.
(c) As
the trait is bell-shaped and as the area in
equals the area in
 ,
by symmetry, we have for the interval (-1.74,0):
The percentage
of the population falling within (-1.74,0) = %p(-1.74,0)
%a(-1.74,0) = %a(0,1.74)
= the area depicted within:
We use the
table in
Centrality,
Spreads, Normality, and Chebyshev's Rule
to determine the percentage. We go down to row 1.7
and over to column 4 to obtain the percentage. The applicable part
of the table appears
below with the applicable cell highlighted.
|
Part of
Table
I: Table of Percentages
Corresponding to Z-scores |
|
|
|
z to one digit to the
right of the decimal point |
z's second digit to the
right of the decimal point |
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
0.0 |
0.0% |
0.4% |
0.8% |
1.2% |
1.6% |
2.0% |
2.4% |
2.8% |
3.2% |
3.6% |
|
|
|
|
|
|
|
|
|
|
|
|
|
1.6 |
44.5% |
44.6% |
44.7% |
44.8% |
44.9% |
45.1% |
45.2% |
45.3% |
45.4% |
45.4% |
|
1.7 |
45.5% |
45.6% |
45.7% |
45.8% |
45.9% |
46.0% |
46.1% |
46.2% |
46.2% |
46.3% |
|
1.8 |
46.4% |
46.5% |
46.6% |
46.6% |
46.7% |
46.8% |
46.9% |
46.9% |
47.0% |
47.1% |
|
|
|
|
|
|
|
|
|
|
|
|
|
3.0 |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
49.9% |
|
Note: The unit
of measure of the z-score is standard deviation. |
|
Sources:
Statistics for Business and Economics: Eighth Edition,
p. 987, and Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, pp. 966-972. |
Hence, the
percentage of the population falling within (-1.74,0)
45.9%.
(d) As
the trait is bell-shaped, we have for the interval (.50,1.67):
%p(.50,1.67)
%a(.50,1.67) = %a(0,1.67)
- %a(0,.50)
and
pictorially,
We use the
table in
Centrality,
Spreads, Normality, and Chebyshev's Rule
to determine the percentages. The applicable parts
of the table appear
below with the applicable cells highlighted.
|
Parts of
Table
I: Table of Percentages
Corresponding to Z-scores |
|
|
|
z to one digit to the
right of the decimal point |
z's second digit to the
right of the decimal point |
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
0.0 |
0.0% |
0.4% |
0.8% |
1.2% |
1.6% |
2.0% |
2.4% |
2.8% |
3.2% |
3.6% |
|
|
|
|
|
|
|
|
|
|
|
|
|
0.4 |
15.5% |
15.9% |
16.3% |
16.6% |
17.0% |
17.4% |
17.7% |
18.1% |
18.4% |
18.8% |
|
Statistics
101
