1
BES Tutorial Sample Solutions, S1/10
WEEK 8 TUTORIAL EXERCISES (To be discussed in the week starting
April 26)
1.
A machine produces sausages with a mean length of 15cm and a standard
deviation of 1.5cm.
A random sample is taken of 36 sausages made by the
machine.
What is the probability that the sample mean length of these
sausages is greater than 14.8cm?
Let
=
X
length of sausage
X
∼
)
)
5
.
1
(
,
15
?(
2
Since
n=36
is
large
we
can
invoke
the
central
limit
theorem
X
∼
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
36
)
5
.
1
(
,
15
2
N
approximately
Then
7881
.
0
2881
.
0
5
.
0
)
8
.
0
0
(
5
.
0
)
8
.
0
(
)
6
5
.
1
(
15
8
.
14
)
8
.
14
(
=
+
=
<
<
+
=
−
>
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
>
=
>
Z
P
Z
P
Z
P
X
P
2.
Suppose a normally distributed random variable
X
has a mean of 50 and a
variance of 100.
Also suppose a sample of size 16 is drawn from this
population.
Calculate the following probabilities:
(a)
P(
X
> 55)
3085
.
0
1915
.
0
5
.
0
)
5
.
0
(
10
50
55
)
55
(
=
−
=
>
=
⎟
⎠
⎞
⎜
⎝
⎛
−
>
=
>
Z
P
Z
P
X
P
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2
(b)
X
~
)
16
100
,
50
(
N
0228
.
0
4772
.
0
5
.
0
)
2
0
(
5
.
0
)
2
(
4
10
50
55
)
55
(
=
−
=
<
<
−
=
>
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
>
=
>
Z
P
Z
P
Z
P
X
P
(c)
)
55
40
(
<
<
X
P
5328
..
0
1915
.
0
3413
..
0
)
5
.
0
0
(
)
1
0
(
)
5
.
0
1
(
10
50
55
10
50
40
)
55
40
(
=
+
=
<
<
+
<
<
=
<
<
−
=
⎟
⎠
⎞
⎜
⎝
⎛
−
<
<
−
=
<
<
Z
P
Z
P
Z
P
Z
P
X
P
(d)
)
55
40
(
<
<
X
P
9772
.
0
4772
.
0
5
...
0
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 Three '11
 DenzilGFiebig
 Statistics, Normal Distribution, Standard Deviation, SAMPLE SIZE

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