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Chapter 7
Sampling and Sampling
Distributions
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Simple Random Sample
Suppose that we want to select a sample of
n objects from a population of N objects.
A simple random sample is selected such
that every object has an equal probability
of being selected and the objects are
selected independently - -the selection of
one object does not change the probability
of selecting any other objects.
Simple Random Sample
Simple random samples are the ideal
sample. In a number of real world
sampling studies analysts develop
alternative sampling procedures to lower
the costs of sampling. But the basis for
determining if these strategies are
acceptable is to determine how closely they
approximate a simple random sample.
Sampling Distributions
Consider a random sample selected from a population to
make an inference about some population characteristic,
such as the population mean, by using a sample statistic such
as a sample mean, X. The inference is based on the
realization that every random sample would have a different
number for X and thus X is a random variable. The
sampling distribution of this statistic is the probability
distribution of the values it could take over all possible
samples of the same number of observations drawn from the
population.
Sampling Distributions
Consider a random sample selected from a population
to make an inference about some population
characteristic, such as the population mean, by using a
sample statistic such as a sample mean, X. The
inference is based on the realization that every random
sample would have a different number for X and thus
X is a random variable. The sampling distribution of
this statistic is the probability distribution of the values
it could take over all possible samples of the same
number of observations drawn from the population.
Sample Mean
Let X1, X2, . . . Xn be a random sample from a
population. The sample mean value of these
observations is defined as
1 n
X   Xi
n i 1
Results for the Sampling
Distribution of the Sample Mean


Let X denote the sample mean of a random sample
of n observations from a population with a mean X
and variance 2. Then
The sampling distribution of X has mean
E (X )  
The sampling distribution of X has standard
deviation
X 

n
This is called the standard error of X.
Results for the Sampling
Distribution of the Sample Mean
3.
If the sample size is not small compared to the
population size N, then the standard error of X is

N n
X 

N 1
n
4.
If the population distribution is normal, then the
random variable
X 
z
X
Has a standard normal distribution with mean 0 and
variance 1.
Standard Normal Distribution for
the Sample Mean
Whenever the sampling distribution of the
sample mean is a normal distribution we can
compute a standardized normal random variable,
Z, that has mean 0 and variance 1
Z
X 
X

X 

n
Central Limit Theorem
Let X1, X2, . . . , Xn be a set of n independent random
variables having identical distributions with mean 
and variance 2, with X as the sum and X as the mean
of these random variables. As n becomes large, the
central limit theorem states that the distribution of
Z
X  X
X

X  n X
n
2
approaches the standard normal distribution.
Sample Proportions
Let X be the number of successes in a binomial sample of n
observations, with parameter . The parameter  is the
proportion of the population members that have a characteristic
of interest. We define the sample proportion as
X
p
n
The sum X is the sum of a set of n independent Bernoulli
random variables each with a probability of success . As a
result p is the mean of a set of independent random variables
and the results developed in the previous sections for sample
means apply. In addition the central limit theorem can be used
to argue that the probability distribution for p can be modeled
as a normal.
Sampling Distribution of the
Sample Proportion
Let p denote the sample proportion of successes in a random sample from a
population with proportion of success . Then
1.
The sampling distribution of p has mean 
E ( p)  
2.
The sampling distribution of p has standard deviation
p 
3.
 (1   )
n
If the sample size is large, the random variable
Z
p 
p
is approximately distributed as a standard normal. The approximation is good
if
n (1   )  9.
Sample Variance
Let X1, X2, . . . , Xn be a random sample from a
population. The quantity
n
1
2
2
s 
(Xi  X )

n  1 i 1
Is called the sample variance and its square root s is
called the sample standard deviation. Given a specific
random sample we would compute the sample
variance and the sample variance would be different
for each random sample, because of differences in
sample observations.
Sampling Distribution of the Sample
Variances
Let s2X denote the sample variance for a random sample of n
observations from a population with variance 2. Then
1.
The sampling distribution of s2 has mean 2
E (s 2 )   2
2.
The variance of the sampling distribution of s2X depends on the
underlying population distribution. If that distribution is normal,
then
4
2

2
Var ( s ) 
n 1
3.
If the population distribution is normal then (n-1)s2/ 2 is
distributed as 2(n-1)