Day 1

Welcome to STAT 240!

About me
About you!

About this course

Admin Stuff

Website
Office Hours and Help Lab
Syllabus story time
“Extra” Math help course

Philosophy

Gen Eds
Pedagogy
Grades

Goals

Statistical Literacy
Basic Analysis
(Hopefully) Probabilistic Thinking

Questions?

Goals For Today:

Develop a definition of the science of Statistics
Define and describe populations versus samples
Define fundamental sampling techniques

Visualizing Data

Statistics as a Science

How many undergraduates are in this room?

How many undergraduates are at K-State?

How many undergraduates are there in America? The world?

I want to understand the average caloric intake of undergraduate students in the US.

But I live in Kansas, not the entirety of the US.

How can I find the answer to my question (assuming we don’t know the answer already)?

I resolve to take my research question:

What is the average caloric intake of undergraduate students in the US?

And hand it over to a 5 different Universities across the US:

Kansas State University
UC Davis
(The) Ohio State University
UCONN (University of Conneticuit)
Texas A&M

I tell them to select \(200\) students at random and determine their caloric intake.

All in all, I end up with \(1000\) students representing the totality of American Undergraduates.

What have I done? I’ve taken a sample from my population.

Population: the entire collection of individuals about which information is sought.
Sample: a subset of population, containing the individuals that are actually observed.

I calculate an average from my sample, and I end up finding out some value that I use to infer the caloric intake for undergraduates across the entire US.

What have I done? I’ve made inference about a population from a sample; I’ve done Statistics.

Statistics: is the study of procedures for collecting, describing, and drawing conclusions from information.

Plainly, the act of describing or making inferences about a population, from a sample

Parameters and Statistics

Let’s look at a different example

Raccoons get rabies, more than normal for most mammals. The Kansas Department of Wildlife & Parks decides to investigate how prevalent rabies is in the state.

KDWP estimates there are roughly \(3.3\) million raccoons in Kansas. They capture \(10000\) raccoons across the state and test them for rabies. They find 382 raccoons that test positive for rabies, with the rest being negative.

In this study, what is considered the population?
What is the sample?
What does the study tell us about raccoons and rabies in Kansas?

In our study, we had a distinct population and sample, with a distinct quantity for each. This number can be very useful, but is generally insufficiently informative.

The Center for Disease Control (CDC) estimates roughly \(10\)-\(14\%\) of raccoons carry Rabies lyssavirus. From KDWP’s study, they found that \(3.82\%\) of the raccoons in their sample had rabies, and extrpolated that to the entire population.

\[{382\over 10000}=0.0382 \times 100 = 3.82\%\]

What we now have are two values that describe our population and our sample.

Parameter: a number that describes an entire population.
Statistic: a number that describes a sample.

What was our parameter in the above study?
Our statistic?

Sampling Techniques

I want to know how many individuals in the K-State Division of Biology consider Cell Biology to be an enjoyable class.

I decide to assign every student who’s taken the class and declared a major that falls under the Division of Biology’s umbrella a number from \(1\) to \(500\) (let’s play some make believe on how many students we’re working with here).

I then generate \(50\) random numbers from \(1\) to \(N\), and select those students to participate in my one question survey.

What I’ve done has resulted in a sample size of \(n=50\), where every individual was equally likely to be selected.

\[\text{Probability of being selected first} = {1\over 500}\]

Let’s not discuss what happens as each person is selected…

I’ve performed a Simple Random Sample.

Simple Random Sample (SRS): a sample chosen by a method in which collection of n population items is equally likely to make up the sample.

Let’s say I divide the \(500\) students into two groups: Pre-Med and Not-Pre-Med

I end up with a split of \(n_1=300\) Pre-Med students and \(n_2=200\) Not-Pre-Med students, then I perform my SRS on each group.

This is a Stratified Sample.

Stratified Sample: The population is divided into groups, called strata, where the members of each stratum are similar in some way. Then a SRS is drawn from each stratum.

Let’s go back to our raccoon example:

KDWP samples those raccoons from pre-defined areas, subsections of Kansas, rather than going across the entirety of Kansas in a big fire line and snatching up suspicious racoons.

Each of those subsections of land are called clusters, and the technique we’ve used here is called Cluster Sampling.

Cluster Sampling: Items are drawn from the population in groups, or clusters.

I drive a Honda Fit, which was built on an assembly line, and part of the process of building that car on an assembly line was something called quality assurance.

Considering my car has yet to blow up on me, it seems to have passed the QA check.

Likely because they used a proper sampling technique:

The part of the assembly line that produces the muffler for Honda Fits decides that every day they’ll draw a number, \(k\), between \(3\) and \(6\).

Whatever number they draw is now the first muffler that comes off of the line to be checked for defects. Then they’ll check the \(k^{th}\) item moving forward.

They draw the number \(4\). So the \(4^{th}\) muffler that comes off of the line will be evaluated for possible defects, then every \(4\) mufflers after that will also be checked for defects.

This is called Systematic Sampling.

Systematic Sampling: a starting point is chosen randomly and then every \(k^{th}\) item in the population is selected.

Recall our example with Undergraduate caloric intake:

I instead instruct the \(5\) chosen Universities to send the survey to every student’s email inbox. They are given the option on whether or not to participate.

This is naturally called a Voluntary Response Sample, due to the participants getting to choose whether they are involved or not.

Let’s say I decide to calculate the caloric intake of Undergraduates in the US based off of the students in this classroom.

That’s called a Sample of Convenience. It was easy, I could finalize it right now, and it’d be fairly incorrect.