Download Sampling Distribution of the Sample Mean

Sampling Distribution of the Sample Mean
Consider the population described by the probability distribution shown
below:
x
1
2
3
4
5
P(x) 0.2 0.3 0.2 0.2 0.1
The expected value of X µ X = E(X) = 2.70 and the standard deviation σ X =
V (X ) = 1.27
Now, take a random sample of size 2 independently from this population
(with replacement), the probability distribution of the sample mean X are
shown below:
Sample
1, 1
1, 2
1, 3
1, 4
1, 5
2, 1
2, 2
2, 3
2, 4
2, 5
3, 1
3, 2
3, 3
3, 4
3, 5
4, 1
4, 2
4, 3
4, 4
4, 5
5, 1
5, 2
5, 3
5, 4
5, 5
X
1
1.5
2
2.5
3
1.5
2
2.5
3
3.5
2
2.5
3
3.5
4
2.5
3
3.5
4
4.5
3
3.5
4
4.5
5
Probability
0.04
0.06
0.04
0.04
0.02
0.06
0.09
0.06
0.06
0.03
0.04
0.06
0.04
0.04
0.02
0.04
0.06
0.04
0.04
0.02
0.02
0.03
0.02
0.02
0.01
The sampling distribution of the sample mean X becomes:
X
p( X )
1
1.5 2
2.5 3
3.5 4
4.5 5
0.04 0.12 0.17 0.20 0.20 0.14 0.08 0.04 0.01
The probability histogram for the sampling distribution of the sample mean
X is shown below:
Sampling Distribution of the Sample Mean
25
Percent
20
15
probability
10
5
0
Sample Mean
The expected value of the sample mean is
µ X = E ( X ) = 2.70 = µ X
and the standard deviation of the sample mean is
σ X = V ( X ) = 0.90 =
σX
n
=
1.27
2