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412
Chapter 8
Sampling Distributions
8.1 Distribution of the Sample Mean
1.
The sampling distribution of a statistic (such as the sample mean) is the probability
distribution for all possible values of the statistic computed from samples of fixed size,
n
.
2.
The Central Limit Theorem states that the sampling distribution of the sample mean,
x
,
becomes increasingly more normal as the sample size
n
increases (provided
X
σ
is finite).
3.
standard error of the mean
4.
zero
5.
The mean of the sampling distribution of
x
is given by
x
μ
=
and the standard deviation
is given by
x
n
=
.
6.
To say that the sampling distribution of
x
is normal, we would require that the population
be normal. That is, the distribution of
X
must be normal.
7.
four; To see this, note that
1
2
4
x
nn
==
⋅
.
8.
True; assuming we are drawing random samples, these two results are true regardless of the
distribution of the population.
9.
The sampling distribution would be exactly normal. The mean and standard deviation
would be
30
x
and
8
2.53
10
x
n
≈
.
10.
No, because the sample size is large, i.e.
40
30
n
=
≥
.
If the distribution of the population
is not normal, the sampling distribution is approximately normal with mean
50
x
and
42
0.63
40
10
x
n
=
=
.
Note:
The answers to the exercises in this section are based on values from the standard
normal table.
Since answers computed using technology do not involve rounding of
Z
scores, they will often differ from the answers given below in the third and fourth
decimal places.
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Distribution of the Sample Mean
413
11.
80
x
μ
==
;
14
14
2
7
49
x
n
σ
=
=
12.
64
x
;
18
18
3
6
36
x
n
=
=
13.
52
x
;
10
2.182
21
x
n
≈
14.
27
x
;
6
1.549
15
x
n
≈
15.
(a)
Since
80
=
and
14
=
, the mean and standard deviation of the sampling
distribution of
x
are given by:
80
x
;
14
14
2
7
49
x
n
=
=
We are not told that the population is normally distributed, but we do have a large
sample size (
30
n
≥
). Therefore, we can use the Central Limit Theorem to say that the
sampling distribution of
x
is approximately normal.
(b)
()
83 80
83
1.50
1
1.50
1 0.9332
0.0668
2
Px
PZ
−
⎛⎞
>=
>
=
>
=
−
≤
=
−
=
⎜⎟
⎝⎠
(c)
75.8 80
75.8
2.10
0.0179
2
−
≤=≤
=
≤
−
=
(d)
78.3 80
85.1 80
78.3
85.1
0.85
2.55
22
0.9946 0.1977
0.7969
P
Z
P
Z
−−
<<
=
= −
=−=
16. (a)
Since
64
=
and
18
=
, the mean and standard deviation of the sampling
distribution of
x
are given by:
64
x
;
18
18
3
6
36
x
n
=
=
We are not told that the population is normally distributed, but we do have a large
sample size (
30
n
≥
). Therefore, we can use the Central Limit Theorem to say that the
sampling distribution of
x
is approximately normal.
(b)
62.6 64
62.6
0.47
0.3192
3
−
<=<
=
<
−
=
(c)
68.7 64
68.7
1.57
1
1.57
3
1 0.9418
0.0582
−
≥=
≥
=
−
<
=−
=
Chapter 8
Sampling Distributions
414
(d)
()
59.8 64
65.9 64
59.8
65.9
1.40
0.63
33
0.7357 0.0808
0.6549
Px
P
x
P
Z
−−
⎛⎞
<<
=
= −
⎜⎟
⎝⎠
=−=
17. (a)
The population must be normally distributed. If this is the case, then the sampling
distribution of
x
is exactly normal. The mean and standard deviation of the sampling
distribution are
64
x
μ
==
and
17
4.907
12
x
n
σ
≈
.
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This note was uploaded on 11/21/2011 for the course NGN 111 taught by Professor Ahmad during the Spring '11 term at American Dubai.
 Spring '11
 ahmad

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