1
Chapter 8

Sampling Distributions
Answers
Section 8.1

Distribution of the Sample Mean
Statistics such as
x
are random variables since their value varies from sample to sample.
As
such, they have probability distributions associated with them.
Sampling Distribution of a Statistic
is a probability distribution for all possible values of
the statistic computed from a sample of size
n
.
Example:
The following data represent the travel times (in minutes) to work for all seven
employees of a startup web development company.
23, 36, 31, 18, 5, 26, 43
Population Mean:
μ
= 26
Samples of size 3:
5, 18, 23
5, 23, 31
18, 26, 36
31, 36, 43
Sample means:
x
= 15.7
x
= 19.7
x
= 26.7
x
= 36.7
Sampling Distribution of the Sample Mean,
x
, is a probability distribution of all possible
values of the sample mean computed from a sample of size
n
from a population with mean μ
and standard deviation
.
Properties of the sampling distribution of the mean:
1.
The sample means target the value of the population mean.
(That is, the mean of
the sample means is the population mean.
The expected value of the sample
mean is equal to the population mean.)
2.
The distribution of sample means tends to be a normal distribution.
(The
distribution tends to become closer to a normal distribution as the sample size
increases.)
3.
As the size of the sample increases, the standard deviation of the distribution of
the sample mean decreases.
The Mean and Standard Deviation of the Sampling Distribution of
x
:
Suppose that a simple random sample of size
n
is drawn from a large population with mean
μ and standard deviation
.
The sampling distribution of
x
will have mean
x
=
,
standard deviation
x
n
=
.
The standard deviation of the sampling distribution of
x
,
X
, is called the
standard error
of the mean
.