Chapter 5: Sampling Distributions
I. Introduction (IPS pages366-367)
A. The distribution of a statistic – A statistic from a random sample or randomized
experiment is a random variable. The probability distribution of the statistic is its
sampling distribution.
B. Population Distribution – The population distribution of a variable is the distribution of its
values for all members of the population. The population distribution is also the
probability distribution of the variable when we choose one individual from the population
at random.
II. Sampling Distributions for Counts and Proportions (IPS section 5.1 pages 367-391)
NOTE: we will use  as the symbol for population proportion and p for sample proportion!!
Although, the symbol will usually have ‘words’ with it, be sure you know the difference since
this is different from your textbook.
A. The binomial distributions for sample counts (any kind of categorical data can be
represented by the binomial distribution --- a “success” means an observation is in the
desired category, “failure” means it is not in the desired category)
1. The Binomial Setting
a. There are a fixed number n of observations.
b. The n observations are all independent.
c. Each observation fall into one of just two categories, which for convenience
we call “success” and “failure.”
d. The probability of a success, call it , is the same for each observation.
2. Binomial Distribution – The distribution of the count X of successes in the
binomial setting is called the binomial distribution with parameters n and . The
parameter n is the number of observations, and p is the probability of a success
on any one observation. The possible values of X are the whole numbers from
0 to n. As an abbreviation, we say that X ~ B(n,) meaning X is distributed as a
binomial random variable with n observations and fixed probability of success, .
B. Sampling Distribution of a Count – When the population is much larger than the sample,
the count X of successes in an SRS of size n has approximately the B(n,) distribution if
the population proportion of successes is . As a rule of thumb, we will use the binomial
sampling distribution for counts when the population is at least 10 times as large as the
sample.
C. Binomial Mean and Standard Deviation – If a count X has the binomial distribution B
(n,) then
μ X = n
σX = n (1   )
D. Sample Proportions:
count of successes in sample
size of sample
X

n
p
E. Mean and Standard Deviation of a Sample Proportion
 pˆ  
 pˆ 
 (1   )
n
Moore, David and McCabe, George. 2002. Introduction to the Practice of Statistics. W. H. Freeman and
Company, New York. 365-413.
F. Normal Approximation for Counts and Proportions – Draw a SRS of size N from a large
population having population proportion  of successes. Let X be the count of
successes in the sample and p  X / n the sample proportion of successes. When n is
large, the sampling distributions of these statistics are
approximately normal:
X is approximately N (n , n (1   )
(
p is approximately N  ,
 (1   )
n
)
As a rule of thumb, we will use this approximation for values of n and  that satisfy n ≥
10 and n(1-) ≥ 10.
III. The Sampling Distribution of a Sample Mean (IPS section 5.2 pages 391-413)
A. Mean and Standard Deviation of a Sample Mean – Let X be the mean of an SRS of
size n from a population having mean  and standard deviation  . The mean and
standard deviation of X are
X  

X 
n


B. Sampling Distribution of a Sample Mean – If a population has the N  ,  X2 distribution.
C. Central Limit Theorem – Draw a SRS of size n from any population with mean  and
finite standard deviation  . When n is large, the sampling distribution of the sample
mean x is approximately normal:

σ 2X 
X is approximately N  μ X , 
n 

FIGURE 5.10 The central limit
theorem in action: the distribution of
sample means from a strongly
nonnormal population becomes more
normal as the sample size increases.
(a) The distribution of 1 observation.
(b) The distribution of x for 2
observations. (c) The distribution of
x for 10 observations. (d) The
distribution of x for 25 observations.
Moore, David and McCabe, George. 2002. Introduction to the Practice of Statistics. W. H. Freeman and
Company, New York. 365-413.