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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 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 Distribution of the Sample
Means from the Worker Population
(Table 7.2)
Sample Mean X
Probability of X
4.50
1/15
4.75
2/15
5.00
2/15
5.25
2/15
5.50
1/15
5.75
3/15
6.00
1/15
6.25
2/15
6.75
1/15
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
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
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
1.
The sampling distribution of X has mean
2.
E (X )  
The sampling distribution of X has standard deviation
X 
3.

n
This is called the standard error of X.
If the sample size is not small compared to the population size N,
then the standard error of X is

4.
N n
X 

N 1
n
If the population distribution is normal, then the random variable
z
X 
X
Has a standard normal distribution with mean 0 and variance 1.
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)
Key Words
 Central Limit Theorem
 Chi-Square Distribution
 Sample Mean
 Sample Proportion
 Sample Variance
 Sampling Distribution
of the Sample Mean
 Sampling Distribution of
the Sample Proportion
 Sampling Distribution of
the Sample Variance
 Simple Random Sample
 Standard Normal for
Sample Mean
 Statistics and Sampling
Distribution