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PSTAT 120C:
Assignment # 7
Due June 1, 2006
Please turn in this assignment at the beginning of lecture next Thursday.
1. We want to ﬁnd the expected value of the
F
statistic under the alternative hypothesis in the
oneway ANOVA. Suppose that each
Y
i,j
∼ N
(
μ
i
, σ
2
) and they are all independent. Let ¯
y
i
be
the mean of the
n
i
observations in the
i
th group. Let ¯
y
be the mean of all
n
observations.
(a) Show that
¯
y
i

¯
y
=
1
n
k
X
`
=1
n
`
(¯
y
i

¯
y
`
)
and then that
¯
y
i

¯
y
=
±
n

n
i
n
²
¯
y
i

1
n
X
`
6
=
i
n
`
¯
y
`
.
(b) Use the result from question 1a) to show that
Var(¯
y
i

¯
y
) =
±
n

n
i
n
²
σ
2
n
i
.
(c) The numerator of the
F
statistic is
1
k

1
k
X
i
=1
n
i
(¯
y
i

¯
y
)
2
.
Calculate its expected value in terms of
σ
2
,
μ
=
E
(¯
y
), and the
k
means
μ
i
=
E
(¯
y
i
).
(Note:
μ
=
1
n
∑
n
i
μ
i
.)
(d) Use the answer from the above calculation to calculate the expectation of the
F
statistic.
2. Complete these ANOVA tables and decide whether or not you would reject the null hypothesis
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This note was uploaded on 04/09/2008 for the course PSTAT 120c taught by Professor Idk during the Spring '07 term at UCSB.
 Spring '07
 idk

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