the population. Select three random samples of 5
days. Calculate the mean rooms rented for each
sample and compare to the population mean. What
is the sampling error in each case?
Population mean,
𝜇 =
Σ𝑥
𝑁
=
0+2+3…+3
30
= 3.13

Example: Sampling Error
Random sample #1
:
4, 7, 4, 3 and 1
𝑥
1
=
Σ𝑥
𝑁
=
4 + 7 + 4 + 3 + 1
5
= 3.8
Sampling error #1,
𝑥
1
− 𝜇
= 3.8 − 3.13 = 0.67
Random sample #2
:
3, 3, 2, 3 and 6
𝑥
2
=
Σ𝑥
𝑁
=
3 + 3 + 2 + 3 + 6
5
= 3.40
Sampling error #2,
𝑥
2
− 𝜇
= 3.4 − 3.13 = 0.2

Example: Sampling Error
Random sample #3
:
0, 0, 3, 3 and 3
𝑥
3
=
Σ𝑥
𝑁
=
0 + 0 + 3 + 3 + 3
5
= 1.8
Sampling error #3,
𝑥
3
− 𝜇
= 1.8 − 3.13 = −1.33
+ve error indicate that the sample mean overestimate the
population mean, and
–
ve error underestimate the
population mean
Possible combination,
𝑛
𝐶
𝑟
=
30!
5! 25!
= 142506
Sampling error for all 142,506 samples
= 0
➔
unbiased

Sampling Distribution of Sample Mean
▪
How to determine
how accurate the sample mean
is?
SAMPLING DISTRIBUTION OF THE SAMPLE MEAN
A probability distribution of all possible sample
means of a given sample size
▪
For a given sample size, the
mean of all possible
sample means selected from a population is equal to
the population mean
▪
There is less variation in the distribution of the sample
mean than in the population distribution
▪
The sampling distribution of the sample mean tends to
become bell-shaped

Example: Sampling Distribution
Tartus Industries has 7 production employees (the population).
The hourly earnings of each employee is as below:
Employee
Hourly Earnings
Employee
Hourly Earnings
Joe
$14
Jan
14
Sam
14
Art
16
Sue
16
Ted
18
Bob
16
1.
What is the population mean?
2.
What is the sampling distribution of the sample mean for
samples of size 2?
3.
What is the mean of the sampling distribution?
4.
What observations can be made about the population
and the sampling distribution?

Example: Sampling Distribution
1.
What is the population mean?
𝜇 =
Σ𝑥
𝑁
=
$14+14+16+16+14+16+18
7
= $15.43
2.
What is the sampling distribution of the sample
mean for samples of size 2?
7
𝐶
2
=
7!
2! 5!
= 21
The 21 sample means from all possible samples of 2
are shown in the table. These 21 sample means are
used to construct a probability distribution

Sampling Distribution Example
Sample
Employees
Hourly
Earnings
Sum
Mean
Sample
Employees
Hourly
Earnings
Sum
Mean
1
Joe, Sam
$14,$14
$28
$14
12
Sue, Bob
16,16
32
16
2
Joe, Sue
14,16
30
15
13
Sue, Jan
16,14
30
15
3
Joe, Bob
14,16
30
15
14
Sue, Art
16,16
32
16
4
Joe, Jan
14,14
28
14
15
Sue, Ted
16,18
34
17
5
Joe, Art
14,16
30
15
16
Bob, Jan
16,14
30
15
6
Joe, Ted
14,18
32
16
17
Bob, Art
16,16
32
16
7
Sam, Sue
14,16
30
15
18
Bob, Ted
16,18
34
17
8
Sam, Bob
14,16
30
15
19
Jan, Art
14,16
30
15
9
Sam, Jan
14,14
28
14
20
Jan, Ted
14,18
32
16
10
Sam, Art
14,16
30
15
21
Art, Ted
16,18
34
17
11
Sam, Ted
14,16
32
16

Sampling Distribution Example
Sample Mean
Number of Means
Probability
$14
3
.1429
15
9
.4285
16
6
.2857
17
3
.1429
21
1.0000
3.
What is the mean of the sampling distribution?
𝜇
ҧ𝑥
=
Sum of all sample means
Total # samples
=
$14+15+15+⋯+16+17
21
= $15.43