362
Chapter 7
The Normal Probability Distribution
7.1
Properties of the Normal Distribution
1.
For the graph to be that of a probability density function,
(1) The area under the graph over all possible values of the random variable must equal 1;
(2) The graph must be greater than or equal to 0 for all possible values of the random
variable.
That is, the graph of the equation must lie on or above the horizontal axis for
all possible values of the random variable.
2.
distribution; density
3.
The area under the graph of a probability density function can be interpreted as either:
(1) The proportion of the population with the characteristic described by the interval; or
(2) The probability that a randomly selected individual from the population has the
characteristic described by the interval.
4.
We standardize the normal random variables so that one table can be used to find the area
under the curve of any normal density function.
5.
;
μ
σμσ
−+
6.
As
σ
increases, the height of the graph of the normal density function decreases.
7.
No, the graph cannot represent a normal density function because it is not symmetric.
8.
Yes, the graph can represent a normal density function.
9.
No, the graph cannot represent a normal density function because it crosses below the
horizontal axis.
That is, it is not always greater than or equal to 0.
10.
No, the graph cannot represent a normal density function because it does not approach the
horizontal axis as
X
increases (and decreases) without bound.
11.
Yes, the graph can represent a normal density function.
12.
No, the graph cannot represent a normal density function because it is not bellshaped.
13.
The figure presents the graph of the density function with the
area we wish to find shaded.
The width of the rectangle is
10 5
5
−=
and the height is
1
30
.
Thus, the area between 5 and
10 is 5
11
30
6
⎛⎞
=
⎜⎟
⎝⎠
.
The probability that the friend is between 5
and 10 minutes late is
1
6
.
03
0
510
1
30
__
X
Random Variable (time)
Density
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Properties of the Normal Distribution
363
14.
The figure presents the graph of the density function with the
area we wish to find shaded.
The width of the rectangle is
25 15 10
−=
and the height is
1
30
.
Thus, the area between 15
and 25 is 10
11
30
3
⎛⎞
=
⎜⎟
⎝⎠
.
The probability that the friend is between
15 and 25 minutes late is
1
3
.
03
0
25
15
1
30
__
X
Random Variable (time)
Density
15.
The figure presents the graph of the density function with the
area we wish to find shaded.
The width of the rectangle is
30 20 10
and the height is
1
30
.
Thus, the area between 20
and 30 is 10
30
3
=
.
The probability that the friend is at least
20 minutes late is
1
3
.
0
20
1
30
__
X
Random Variable (time)
16.
The figure presents the graph of the density function with the
area we wish to find shaded.
The width of the rectangle is
505
and the height is
1
30
.
Thus, the area between 0 and 5
is 5
30
6
=
.
The probability that the friend is no more than 5
minutes late is
1
6
.
0
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 Spring '11
 ahmad
 Normal Distribution

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