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Statistics in Applied Science and
Technology
Chapter 7 - Sampling Distribution of
Means
July, 2000
Guang Jin
Key Concepts in This Chapter
 Distribution
of a population
 Distribution of sample means
 Central limit theorem
 Standard error of the mean
 Z score of a sample mean
 Student’s t distribution
 t score and degree of freedom
July, 2000
Guang Jin
The Distribution of a Population and the
Distribution of its Sample Means
A distribution of sample means is the set of
values of sample means obtained from all possible
samples of the same size (n) from a given
population.
 A distribution of a population includes a set of
intervals and displays their frequency (numbers of
cases or occurrences) in each intervals for that
given population.

July, 2000
Guang Jin
Central Limit Theorem
 The
central limit theorem states that for a
randomly selected sample of size n (n25,
but the larger n is, the better the
approximation) with a mean of  and
standard deviation :
• The distribution of sample means x is
approximately normal regardless of whether the
population distribution is normal or not
July, 2000
Guang Jin
Central Limit Theorem (Cont’d)
• The mean of the distribution of sample means is equal
to the mean of the population distribution - that is,
x  
• The standard deviation of the distribution of sample
means is equal to the standard deviation of the
population () divided by the square root of the sample
size (n), that is,

x 
July, 2000
Guang Jin
n
Standard Error of the Mean
 The
standard deviation of the sample
means, referred to as the standard error of
the mean, is denoted as SE( x ), that is,
SE( x )   x 


n
SE ( x ) is a rough measure of the average amount
by which sample mean deviate from population
mean (amount of sampling error).
July, 2000
Guang Jin
In practice, the standard error of
the mean is calculated by:
s
sx 
n
 Where:
S - sample standard deviation
s x - standard error of the mean
estimated from a sample
July, 2000
Guang Jin
Z score of a sample mean
Z
score of a sample mean establishes the
relative position of x in a distribution of
sample means and can be calculated by:
x
Z
/ n
July, 2000
Guang Jin
Student’s t distribution
When sample standard deviation is used to
calculate z score of a sample mean, we no longer
have the standard normal distribution, instead we
have so called Student’s t distribution
 t distribution is similar to the standard normal
distribution and approximate standard normal
distribution when sample size exceeds 30.

July, 2000
Guang Jin
t score and degree of freedom

The equation for t score is:
x
t
s/ n

Degree of freedom (df) can be calculated by:
df  n  1
July, 2000
Guang Jin