STAT 350 Solutions to Homework 1 and Homework 2
19.
(a) The density curve forms a rectangle over the interval [4, 6].
For this reason, uniform
densities are also called
rectangular densities
by some authors.
Areas under uniform
densities are easy to find (i.e., no calculus is needed) since they are just areas of rectangles.
For example, the total area under this density curve is
)
4
6
(
2
1
= 1.
height = 1/(64)
= 1/2
4
6
x
(b)
The proportion of values between 4.5 and 5.5 is depicted (shaded) in the diagram below.
The area of this rectangle is
)
5
.
4
5
.
5
(
2
1
= .5.
Similarly, the proportion of
x
values that
exceed 4.5 would be
)
5
.
4
6
(
2
1
= .75.
4
6
x
4.5
5.5
(c)
The median of this distribution is 5 because exactly half the area under this density sits over
the interval [4,5].
(d)
Since 'good' processing times are short ones, we need to find the particular value
0
x
for
which the proportion of the data less than
0
x
equals .10.
That is, the area under the
density to the left of
0
x
must equal .10.
Therefore, the area
=.10 =
)
4
(
0
2
1
x
, and so
.
20
.
4
0
x
Thus,
.
20
.
4
0
x
21.
(a) The density function is
5
.
12
/
1
)
5
.
7
20
/(
1
)
(
x
f
over the interval [7.5, 20] and
0
)
(
x
f
elsewhere.
The proportion of depths less than
k
is given by the expression
)
5
.
7
(
5
.
12
1
k
for
5
.
7
k
and 0 elsewhere.
For
10
k
, this proportion is
)
5
.
7
10
(
5
.
12
1
= .20.
For
15
k
, it is
)
5
.
7
15
(
5
.
12
1
= .60.
(b)
The proportion of
x
values that are at least
k
is
)
20
(
5
.
12
1
k
.
The proportion of
x
values that
strictly
exceed
k
is also
)
20
(
5
.
12
1
k
because
)
(
x
f
is a continuous density.
For
10
k
, this
proportion is
)
10
20
(
5
.
12
1
= .80; for
15
k
, it is
)
15
20
(
5
.
12
1
= .40.
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(c)
It helps to draw the picture of the density:
7.5
20
x
x
1
x
2
central 90%
So, the area to the right of
1
x
should be .95; i.e.,
)
20
(
1
5
.
12
1
x
=.95.
Similarly, the area to the
right of
2
x
is .05, and so
)
20
(
2
5
.
12
1
x
= .05.
Solving these equations gives
125
.
8
1
x
and
.
375
.
19
2
x
23.
(a)
x
= .00004
0
(b)
dx
e
x
0000
,
20
00004
.
00004
.
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 Spring '08
 Staff
 Statistics, Normal Distribution, 1%, 5%, 15%, 4%, 2%

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