EXERCISES IN STATISTICS
Series A, No. 10
1. A horticulturist considers that a batch of seeds is worth sowing if 50% of the
resulting flowers are going to be pure white. To test the worth of a particular
batch, he sows eight seeds with the intention of sowing the remainder if at least
four of the eight plants have white flowers. Find the probability of his making
a wrong decision (a) if 25% of the seeds are of the white variety, and (b) if 75%
of the seeds are of the white variety.
Answer
Let
p
be the probability that a seed selected at random will produced a
white flower, and let
x
be the number of white flowers in the test sample of
n
= 8
seeds. We may assume that
x
∼
b
(
n, p
) =
n
!
(
n
−
x
)!
x
!
p
x
(1
−
p
)
n
−
x
.
(a) Let
p
= 1
/
4. The wrong decision, which is to sow the seeds, is made if
x
∈
{
4
,
5
,
6
,
7
,
8
}
.
Observe that
P
(
x
∈
{
4
,
5
,
6
,
7
,
8
}
) = 1
−
P
(
x
∈
{
0
,
1
,
2
,
3
,
}
).
Then
P
(
x
∈
{
0
,
1
,
2
,
3
}
) =
3
X
x
=0
n
!
(
n
−
x
)!
x
!
p
x
(1
−
p
)
n
−
x
=
µ
3
4
∂
8
+ 8
µ
1
4
∂ µ
3
4
∂
7
+
8
.
7
2
.
1
µ
1
4
∂
2
µ
3
4
∂
6
+
8
.
7
.
6
3
.
2
.
1
µ
1
4
∂
3
µ
3
4
∂
5
=
µ
1
4
∂
8
n
3
8
+ 8
.
3
7
+ 4
.
7
.
3
6
+ 8
.
7
.
3
5
o
.
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 Spring '12
 D.S.G.Pollock
 Normal Distribution, Null hypothesis, Statistical hypothesis testing, Type I and type II errors, series A, wrong decision

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