Final Practice Problems  Answer Key
1. The following is a fictional joint probability distribution for level of household income and the type of
computer a household owns. Computer type is entered in the left most column, and level of income
on the top most row.
Computer Type/Income
15,000
40,000
90,000
No Computer
0.2
0.1
0
PC
0.1
0.2
0.1
Mac
0
0.1
0.2
(a) What is the probability that household income is 15,000? What is the probability household
income is 40,000? What is the probability that household income is 90,000?
Let
I
denote income,
C
type of computer
NC
no computer and
M
denote Mac. We then obtain:
P
(
I
= 15
,
000) =
P
(
I
= 15
,
000;
C
=
NC
)+
P
(
I
= 40
,
000;
C
=
PC
)+
P
(
I
= 90
,
000;
C
=
M
) = 0
.
3
Similarly, we obtain
P
(
I
= 40
,
000) = 0
.
4 and
P
(
I
= 90
,
000) = 0
.
3.
(b) What is the probability that household income is higher than 30,000?
Since
P
(
I
≥
30
,
000) =
P
(
I
= 40
,
000) +
P
(
I
= 90
,
000) from (a) we obtain:
P
(
I
≥
30
,
000) = 0
.
4 + 0
.
3 = 0
.
7
(c) Conditional on income being 90,000, what is the probability that a household owns a PC?
Using our answer from (a) we quickly obtain:
P
(
C
=
PC

I
= 90
,
000) =
P
(
C
=
PC
;
I
= 90
,
000)
P
(
I
= 90
,
000)
=
1
3
(d) Conditional on income being 90,000, what is the probability that a household owns a Mac?
Following (d) we obtain
P
(
C
=
M

I
= 90
,
000) = 2
/
3.
(e) Are the level of income and computer ownership independent from each other?
They are not. It is easy to verify, for example, by noting that:
P
(
I
= 90
,
000;
C
=
NC
) = 0
6
= 0
.
3
×
0
.
3 =
P
(
I
= 90
,
000)
P
(
C
=
NC
)
2. Suppose
X
is normally distributed with unknown mean
μ
and variance
σ
2
. You sample 25 observa
tions and find
¯
X
= 5 and
s
2
X
= 16.
(a) Construct a 95% confidence interval for
μ
using a tdistribution.
What is the length of your
confidence interval?
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 Spring '10
 Elliot
 Normal Distribution, Normal approximation

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