Sampling Distributions

Contents:

Sampling Distributions
1. Populations and Samples
2. The Sampling Distribution of the mean ( known)
3. The Sampling Distribution of the mean ( unknown)
4. The Sampling Distribution of the Variance

Populations and Samples

Finite
Population
Population
Infinite
Population

Populations and Samples
Population: A set or collection of all the objects, actual
or conceptual and mainly the set of
numbers, measurements or observations which are
under investigation.
Finite Population : All students in a College
Infinite Population : Total water in the sea or all the
sand particle in sea shore.
Populations are often described by the distributions of
their values, and it is common practice to refer to a
population in terms of its distribution.

Finite Populations
• Finite populations are described by the actual
distribution of its values and infinite populations are
described by corresponding probability distribution or
probability density.
• “Population f(x)” means a population is described
by a frequency distribution, a probability distribution
or a density f(x).

Infinite Population
If a population is infinite it is impossible to observe all
its values, and even if it is finite it may be impractical
or uneconomical to observe it in its entirety. Thus it is
necessary to use a sample.
Sample: A part of population collected for investigation
which needed to be representative of population and
to be large enough to contain all information about
population.

Random Sample (finite population):
• A set of observations X1, X2, …,Xn constitutes a
random sample of size n from a finite
population of size N, if its values are chosen so
that each subset of n of the N elements of the
population has the same probability of being
selected.

Random Sample (infinite Population):
A set of observations X1, X2, …, Xn constitutes a random sample
of size n from the infinite population ƒ(x) if:
1. Each Xi is a random variable whose distribution is given
by ƒ(x)
2. These n random variables are independent.

We consider two types of random sample: those drawn with
replacement and those drawn without replacement.

Sampling with replacement:
• In sampling with replacement, each object
chosen is returned to the population before the
next object is drawn. We define a random
sample of size n drawn with replacement, as
an ordered n-tuple of objects from the
population, repetitions allowed.

The Space of random samples drawn with
replacement:
• If samples of size n are drawn with replacement from a
population of size N, then there are Nn such samples. In any
survey involving sample of size n, each of these should have
same probability of being chosen. This is equivalent to making
a collection of all Nn samples a probability space in which each
sample has probability of being chosen 1/Nn.
• Hence in above example There are 32 = 9 random samples of
size 2 and each of the 9 random sample has probability 1/9 of
being chosen.

Sampling without replacement:
In sampling without replacement, an object chosen is not returned
to the population before the next object is drawn. We define
random sample of size n, drawn without replacement as an
unordered subset of n objects from the population.

The Space of random samples drawn
without replacement:
• If sample of size n are drawn without
replacement from a population of size N, then
N 
there are 
 n
 such samples. The collection of




all random samples drawn without
replacement can be made into a probability
space in which each sample has same chance
of being selected.

Mean and Variance
If X1, X2, …, Xn constitute a random sample, then
n

 X i
i 1
X 
n
is called the sample mean and
n

 (X i  X )
2

i 1

2
S
n 1

is called the sample variance.

Sampling distribution:
• The probability distribution of a random variable
defined on a space of random samples is called a
sampling distribution.

The Sampling Distribution of the Mean ( Known)

Suppose that a random sample of n observations has been taken from
some population and x has been computed, say, to estimate the mean
of the population. If we take a second sample of size n from this
population we get some different value for x . Similarly if we take
several more samples and calculate x , probably no two of the x ' s .
would be alike. The difference among such x ' s are generally attributed
to chance and this raises important question concerning their
distribution, specially concerning the extent of their chance of
fluctuations.
Let  X and 
2
X
be mean and variance for sampling distributi on of the
mean X .


2
Formula for μ X and σ X :
Theorem 1: If a random sample of size n is taken from a population
having the mean  and the variance 2, then X is a random variable
whose distribution has the mean  .

2

For samples from infinite populations the variance of this distribution is .
n
 N n
2

For samples from a finite population of size N the variance is n  .
N 1

 2
 (for infinite Population )
 n
That is  X   and   
2
X
 N n
2
 (for finite Population )
 n N 1



Proof of    for infinite population for the
X

continuous
case: From the definition we have
  

X    .....  x f ( x1 , x 2 ,...., x n )dx 1 dx 2 .... dx n
 
   n
xi
   .....   n
f ( x1 , x 2 ,...., x n )dx 1 dx 2 .... dx n
   i 1

n   
1

n
   .....  x i
f ( x1 , x 2 ,...., x n )dx 1 dx 2 .... dx n
i 1     

The Sampling Distribution of the Mean ( Known

Where f(x1, x2, … , xn) is the joint density function of the random variables
which constitute the random sample. From the assumption of random sample
(for infinite population) each Xi is a random variable whose density function
is given by f(x) and these n random variables are independent, we can write
f(x1, x2, … , xn) = f(x1) f(x2) …… f(xn)
and we have
n   
1
X 
n
   .....  x i
f ( x1 ) f ( x 2 ).... f ( x n )dx 1 dx 2 .... dx n
i 1     

n   
1

n
  f ( x1 ) dx 1 ...  x i f ( x i ) dx i ....  f ( x n ) dx n
i 1    


Since each integral except the one with the integrand xi f(xi) equals 1 and the
one with the integrand xi f(xi) equals to , so we will have
n
1 1
X 
n
 
n
n   .
i 1

Note: for the discrete case the proof follows the same steps, with integral sign
replaced by  ' s.
For the proof of  X   2 / n we require the following result
2

Result: If X is a continuous random variable and Y = X - X , then Y = 0
and hence  Y2   X .
2

Proof: Y = E(Y) = E(X - X) = E(X) - X = 0 and 
Y2 = E[(Y - Y)2] = E [((X - X ) - 0)2] = E[(X - X)2] = X2.


• Regardless of the form of the population distribution, the
distribution of is approximately normal with mean  and
variance 2/n whenever n is large.
• In practice, the normal distribution provides an excellent
X
approximation to the sampling distribution of the mean
for n as small as 25 or 30, with hardly any restrictions on the
shape of the population.
• If the random samples come from a normal population, the
X
sampling distribution of the mean is normal regardless of
the size of the sample.

The Sampling Distribution of the mean
( unknown)
• Application of the theory of previous section requires knowledge of the
population standard deviation  .
• If n is large, this does not pose any problems even when  is unknown, as
it is reasonable in that case to use for it the sample standard deviation s.
• However, when it comes to random variable whose values are given by
very little is known about its exact sampling distribution for

small values of n unless we make the assumption that the sample comes
from a normal population.

X  
,
S / n

( unknown)
Theorem : If is the mean of a random sample of size n taken from a
normal population having the mean  and the variance 2, and
X

(Xi  X )
n 2

  , then
2
S
i 1 n 1
X 
t
S/ n
is a random variable having the t distribution with the parameter  = n – 1.
This theorem is more general than Theorem 6.2 in the sense that it does not
require knowledge of  ; on the other hand, it is less general than Theorem 6.2
in the sense that it requires the assumption of a normal population.

( unknown)
• The t distribution was introduced by William S.Gosset in
1908, who published his scientific paper under the pen name
“Student,” since his company did not permit publication by
employees. That’s why t distribution is also known as the
Student-t distribution, or Student’s t distribution.
• The shape of t distribution is similar to that of a normal
distribution i.e. both are bell-shaped and symmetric about the
mean.
• Like the standard normal distribution, the t distribution has the
mean 0, but its variance depends on the parameter , called the
number of degrees of freedom.

( unknown)

t ( =10)
Normal

t ( =1)

Figure: t distribution and standard normal distributions

( unknown)
• When   , the t distribution approaches the standard
normal distribution i.e. when   , t  z.
• The standard normal distribution provides a good
approximation to the t distribution for samples of size 30 or
more.

Sampling Distributions

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