The Basic Idea of Sampling Distributions

The Dispersion in Sampling Distributions

Problem Situations:
the Sampling Distribution of the Sample Mean

Problem Situations: The Sampling Distribution of the
Sample Proportion

In Conclusion
The Basic Idea of Sampling Distributions
Last week, we looked at four problem situations in which probability plays a central role in using data to solve
problems: Contingency Tables, Expected Value, The Binomial Probability Distribution, and The Normal
Probability Distribution. This week, we will narrow our focus to the Normal Probability Distribution, extending its
usefulness tremendously by applying it to a very special case: Sampling Distributions. We will look at two problem
situations within Sampling Distributions: the Sampling Distribution of the Sample Mean, and the Sampling
Distribution of the Sample Proportion.
Now the idea of sampling distributions is very simple, but, as we will see, it is powerful and very useful. Here’s the
basic idea. Imagine that we are interested in the average income of Keller students; we take a random sample of n =
16 students; and we calculate the mean income reported by the students (after assuring them of confidentiality, and
providing a valid reason for our request so that they respond to it and provide an honest answer). The mean that we
calculate gives us an idea about the mean income of all Keller students. This is good. But there’s a problem: if we
take another sample of n = 16 Keller students, and we compute the mean income for them, the mean will be
somewhat different. In fact, imagine taking a great number of random samples of size n = 16, and computing the
mean for each of them. The good news is that they are likely to be roughly the same, but the bad news is that they
will vary one from another. Which one is right? What is the “true” mean income of the population of Keller
students?
Seeing the Pattern in Sampling Distributions
. The beauty of probability distributions, as we saw last week, is that
while any individual random sample’s mean is unknown (before we calculate it), probability distributions have a
pattern. And the pattern that emerges, when we take our large number of sample means and we create their
distribution, is a normal probability distribution. This distribution, like all distributions, has a mean and a standard
deviation.
The Central Tendency in Sampling Distributions
. Now it turns out, very usefully, that the mean of this new
distribution of sample means is equal to the population mean. We represent this as
, where
represents
the mean of the distribution of sample means, and
represents the mean of the original population. So in our
example, if we look at our distribution of sample means of Keller students’ incomes, the mean of this distribution
will point at, and be very close to, the mean income for the population of all Keller students.
The Dispersion in Sampling