IE 111 Fall 2007
Solutions for Homework Assignment 9
Note: I used Excel to evaluate Normal probabilities and percentiles instead of the
Normal tables.
Because the table’s accuracy is limited in significant digits, my
answers may vary somewhat from those of people using the Tables.
Students should
not be penalized for such discrepancies.
Question 1.
The lower and upper specifications on the diameter of a certain manufactured part are
[0.99 cm to 1.01 cm].
If the diameter of a part falls in this range then it is "good".
Otherwise it is defective.
The actual diameter of parts produced is a random variable
distributed N(1.0 ,
0.000025).
a)
What percentage of parts diameters fall within the specifications?
σ
=0.005
(
29
)
2
(
)
2
(
2
2
005
.
0
1
01
.
1
005
.
0
1
99
.
0
)
01
.
1
99
.
0
(

Φ

Φ
=
<
<

=

<
<

=
<
<
X
P
Z
P
X
P
=0.9772500.022750
=
0.9545
b)
Suppose I change the mean from 1.0 to 1.005.
Now what percentage of parts diamet
ers fall within the specifications?
(
29
)
3
(
)
1
(
1
3
005
.
0
005
.
1
01
.
1
005
.
0
005
.
1
99
.
0
)
01
.
1
99
.
0
(

Φ

Φ
=
<
<

=

<
<

=
<
<
X
P
Z
P
X
P
=
0.841345

0.00135
=
0.84
c)
Suppose the mean is 1.0 as in part a).
Suppose further that I can adjust the standard
deviation
σ
.
What value of
σ
will result in 1% of the parts being defective?
0.003882
2.575829
01
.
0
or tables)
function
Norminv
Excel
(From
2.575829
000
.
1
01
.
1
995
.
0
000
.
1
01
.
1
99
.
0
000
.
1
01
.
1
000
.
1
99
.
0
=
=
=

⇒
=

<
⇒
=

<
<

σ
σ
σ
σ
σ
Z
P
Z
P
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Question 2.
The number of miles my car can go on a gallon of gasoline is Normally distributed with a
mean of 25 miles and a standard deviation of 7 miles.
a)
I currently have exactly one gallon in my tank, and have to travel 30 miles to my des
tination.
What is the probability I make it to my destination before I run out of gas?
Let X = the number of miles I go. P(I make it) = P(X>30)
0.237525
0.762475

1
0.714286)
(
1
7
25
30
1
)
30
(
1
=
=
<

=

<

=
<

=
Z
P
Z
P
X
P
b)
Find the distance D so that I have a 90% chance of traveling D miles before running
out of gas (assuming again that I start with one gallon).
03
.
16
D
tables)
(from
1.28155
7
25
1
.
0
7
25
1
.
0
)
(
9
.
0
)
(
1
=
⇒
=

⇒
=

<
⇒
=
<
⇒
=
<

=
D
D
Z
P
D
X
P
D
X
P
c)
Suppose I can change my mean mileage per gallon by adjusting the carburetor.
What
value should I set the mean to so that I have a 90% chance of making it 30 miles on
one gallon of gas?
38.9709
tables)
(from
1.28155
7
30
1
.
0
7
30
1
.
0
)
30
(
9
.
0
)
30
(
1
=
⇒
=

⇒
=

<
⇒
=
<
⇒
=
<

=
μ
μ
μ
Z
P
X
P
X
P
Question 3.
Steel balls are manufactured for use in ball bearings.
The diameter of a randomly
selected ball is Normally distributed with a mean of 3 millimeters and a standard
deviation of 0.2 millimeters.
a)
If the specifications say that ball diameter must be in the range 3
±
0.5 millimeters in
order to meet quality requirements, what percentage of the balls will meet these re
quirements?
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 Spring '07
 Storer
 Normal Distribution, Poisson Distribution, Standard Deviation, Poisson process

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