PSTAT 120C:
Midterm Solutions
May 20, 2009
1.
(a) For
W
= 99
,
235 the appropriate
U
is
U
= 99
,
235

298(299)
2
= 54
,
684
The expected value of this statistic is
n
1
n
2
/
2 = 49
,
021 so we really should use
U
= 298(329)

54
,
684 = 43
,
358
(b) The standard deviation of the
U
statistic here is
p
n
1
n
2
(
n
1
+
n
2
+ 1)
/
12 =
p
(298)(329)(628)
/
12 = 2
,
265
.
141
Thus the test statistic
Z
=
43
,
358

49
,
021
2
,
265
.
141
=

2
.
50
which gives a
P
value of
P
= 2
P
{
Z <

2
.
50
}
= 2(
.
0062) = 0
.
0124.
(c) The advantage of the Mann–Whitney test is that we do not have to assume that the
observations are normally distributed.
(d) The disadvantage is that the Mann–Whitney test has less power than the
t
test when the
data is normally distributed.
2. A survey asked people to rate their opinion on the economy on a scale from 1 to 10 where 10
meant they were “extremely optimistic”, 1 meant they were “extremely pessimistic”, and 5
indicated that they were “neutral.” Here are the results in the order they were collected:
1
,
5
,
8
,
6
,
4
,
9
,
10
,
5
,
5
We wish to test
H
0
:
μ
= 5
.
3 versus
H
0
:
μ
6
= 5
.
3.
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 Spring '08
 EduardoMontoya
 Statistics, Normal Distribution, Null hypothesis, Statistical hypothesis testing, Statistical significance

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