rejection region: outside 0
±
2
.
17
*
104
/
25 =
±
4
.
426.
Then the power is
P
{
Z <

9
.
426
/
2
.
04
}
+
P
{
Z >

0
.
574
/
2
.
04
}
= 0 +
P
{
Z >

0
.
281
}
= 1

0
.
3897 =
0
.
61.
4(a) ˆ
p
A
= 63
/
220 =
.
2864, ˆ
p
C
= 59
/
80 =
.
7375 and
ˆ
p
= (63 + 59)
/
300 =
.
4067. We can approximate the
binomial with a normal distribution because 220
*
ˆ
p
=
41
.
3
>
5 and 220
*
ˆ
q
= 178
.
7
>
5 and similarly
80
*
ˆ
p
= 37
.
4
>
5 and 80
*
ˆ
q >
5. Then
z
= (
.
7375

.
2864)
/
.
4067
*
.
5933
*
(1
/
220 + 1
/
80) = 7
.
034. Us
ing the normal distribution we get
p <
2
*
2
.
9
*
10

7
is
way smaller than
.
0001 and way smaller than
α
=
.
10.
We strongly reject
p
A
=
p
C
: there is strong evidence
that the locus is linked to some genetic region affect
ing flowering time, the C allele being linked to early
flowering. Assumptions include random sampling of
and independence among plants.
(b) Observations are clearly paired:
each leaf observa
tion is paired to the root observation made on the
same plant.
Therefore, one would use a twosided
pairedsample ttest, provided that the distribution
of difference (leaf expression root expression) is not
too far from a normal distribution. Note: there is no
need to assume that
σ
1
=
σ
2
and no need to assume
normality of gene expression. Only the normality of
the expression difference is needed.
5(a)
false
: we mostly use a tdistribution. The normal
distribution is only used when the variance is known.
true
true
because zquantiles are smaller than tquantiles.
false
: the confidence interval is centered at the sam
ple mean (¯
y
, which is random) while the notrejection
region is centered at the null hypothesis
μ
0
(not ran
dom  known before collecting the data)
true
true
: they both have the same length, which is twice
z
α/
2
*
σ/
√
n
.
Summary of grades:
Frequency
20
40
60
80
100
0
10
20
30
40
●
●
●●
●
●
●
76
83
89
1
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 Fall '08
 Staff
 Statistics, Binomial, Normal Distribution, Variance, 2sided test

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