Chapter 8 – Sampling Distributions
Defn: Sampling error is the error resulting from using a sample to infer a population
characteristic.
Example: We want to estimate the mean amount of Pepsi-Cola in 12-oz. cans coming off an
assembly line by choosing a random sample of 16 cans, and using the sample mean as an
estimate of the mean for the population of cans. Suppose that we choose 100 random samples of
size 16 and compute the sample mean for each of these samples. These 100 values of X will
differ from each other somewhat due to sampling error, but the values should all be close to 12oz.
Defn: For a random variable X, and a given sample size n, the distribution of the variable X ,
i.e., of all possible values of X , is called the sampling distribution of the mean. This
probability distribution is a set of pairs of numbers. In each pair, the first number is a possible
value of the sample mean, and the second number is the probability of obtaining that value of the
mean occur when we select a random sample from the population.
Properties of the Sampling Distribution of the Mean:
1) For samples of size n, the expectation (mean) of
other words,
X  X .
X
, equals the expectation (mean) of X. In
2) The possible values of X cluster closer around the population mean for larger samples than
for smaller samples. In other words, the larger the sample size, the smaller the sampling error.
 X , will be
X


X
In particular, we have
n
In particular, the standard deviation of the sampling distribution of the means,
smaller than the population standard deviation,
X.
,
where n is the sample size.
Example (Continued): To visualize the concept of the sampling distribution of the mean, let us
return to the above example. If we were to measure the amount of Pepsi-Cola in each 12-oz.
can, we would obtain a list of values: 𝑥1 , 𝑥2 , 𝑥3 , …. Here, 𝑥1 is the amount of Pepsi-Cola in the
first can selected; 𝑥2 is the amount of Pepsi-Cola in the second can selected; etc. If we were to
construct a histogram for all of these values, that histogram would represent the population
distribution. Assuming that the distribution of fill amounts is normal, the histogram would be a
bell-shaped curve centered at 𝜇𝑋 = 12 𝑜𝑧. and with spread given by some (relatively) small
amount, such as 𝜎𝑋 = 0.11 𝑜𝑧.
Now, instead, let us consider the average fill amount per can for random samples of 16 cans. We
select a random sample of 16 cans, measure the amount in each can, and calculate the sample
mean. We do this repeatedly, and obtain a list of values: 𝑥̅1 is the mean fill amount for the first
sample of 16 cans; 𝑥̅2 is the mean fill amount for the second sample of 16 cans; 𝑥̅3 is the mean
fill amount for the third sample of 16 cans; etc. If we were to construct a histogram for all of
these values, that histogram would represent the sampling distribution of the mean for samples of
size 16. It would be a bell-shaped curve centered at 𝜇𝑋̅ = 𝜇𝑋 = 12 𝑜𝑧. and with spread given by
𝜎
𝜎𝑋̅ = 𝑋 = 0.0275 𝑜𝑧.
√16
Defn: The standard deviation of the sampling distribution of the mean is called the standard
error of the mean.
Property 2 says that for a given population, and a given random variable defined for the members
of that population, the standard error of the mean is smaller for larger sample sizes. To visualize
what this means, let’s look at an example.
Example: Consider the adult population of the United States. We have an IQ test that has been
developed to assess the intelligence of individuals in this population. The distribution of IQ
scores in the population is normal with mean µ = 100 and standard deviation σ = 15. This means
that if we were to measure the IQ of each adult American, and do a histogram of the data, we
would get a bell-shaped curve centered at µ = 100 and with standard deviation σ = 15.
Now suppose that we select a simple random sample of size n = 4 from the population,
administer the IQ test to each person in the sample, and find the sample mean. If we were to do
this repeatedly, so that we obtain sample mean values for all possible samples of size n = 4, and
do a histogram of those numbers, the histogram would be a bell-shaped curve centered at 𝜇𝑋̅ =
𝜎
𝜇 = 100 and with standard deviation 𝜎𝑋̅ = = 7.5.
√4
Suppose that, instead, we consider all possible samples of size n =100 from the population, and
for each sample, we obtain the sample mean IQ score. If we do a histogram of all of these
numbers, the histogram will be a bell-shaped curve centered at 𝜇𝑋̅ = 𝜇 = 100 and with standard
𝜎
deviation 𝜎𝑋̅ =
= 1.5.
√100
These three curves are shown together in the graph below:
The following theoretical result from probability theory is fundamental for our work in statistical
inference.
The Central Limit Theorem: For large (n  30) sample sizes, the random variable X has an
approximate normal distribution, with mean
other words, the random variable Z 
 X   X and standard deviation  X

X
n
. In
X  X
has an approximate standard normal distribution.
 X 


 n
This theorem holds regardless of the type of population distribution. The population distribution
could be normal; it could be uniform (equally likely outcomes); it could be strongly positively
skewed; it could be strongly negatively skewed. Regardless of the shape of the population
distribution, the sampling distribution of the mean will be approximately normal.
Example: p. 390, Exercise 20
Example: p. 390, Exercise 22
Example: p. 391, Exercise 27
The Sampling Distribution of the Sample Proportion
Assume that we have a (large) population, which is divided into two subpopulations. In one
subpopulation, each member possesses a certain characteristic; in the other subpopulation, each
member does not possess this characteristic. Assume that the proportion of members of the
entire population who possess the characteristic is p.
We select a simple random sample of size n from the population. We are interested in the
proportion of the members of the sample who possess the characteristic of interest. This
proportion is called the sample proportion, denoted by 𝑝̂ .
The Central Limit Theorem tells us that, if the sample size is “large,” then
a) The shape of the sampling distribution of 𝑝̂ is approximately normal, having mean p and
𝑝(1−𝑝)
standard deviation √
𝑛
, provided 𝑛𝑝(1 − 𝑝) ≥ 10.
b) Under the same conditions, the distribution of
𝑍=
is approximately standard normal.
Example: p. 399, Exercise 15.
Example: p. 399, Exercise 21
Example: p. 399, Exercise 23
𝑝̂ − 𝑝
√𝑝(1 − 𝑝)
𝑛