Monthly Archives: March 2014

Type I and Type II error are concepts my students have struggled with year after year. This school year, I decided to do something different in AP Stats.

KNEB news story: "Rabies Outbreak Soars in Goshen County"
(this particular county is very close to where we live in western Nebraska)

The news story above was posted the night before I was slated to introduce Type I and Type II error for the first time. The text of the news story caught my eye because something, literally, didn't add up.

...since February 7th of this year, out of 19 dead skunks collected, 16 tested positive for rabies. They also had 1 red fox test positive for the disease... Mills will be retiring next month and has been at the lab since 1984 and has never seen, even in an outbreak situation, this high of a percentage rate of positive samples at 89.5%...

It is interesting in the news story that the reported percentage, 89.5%, does not match the fraction 16/19 (which is approximately 84.2%). The 89.5% figure is a computational error. The person computing it took included the rabid red fox, despite the fact the red fox is a different species, and also did not increase the denominator by one. The 89.5% comes from the fraction 17/19, which is obviously not valid. (We had a nice class discussion over whether this was a computational error or a purposeful error meant to sensationalize the news story.)

When I first saw this story, I decided to scrap what I had planned for the day in AP Stats and made this worksheet in the planning period I had right before class.

APS Rabies Outbreak 3-7-14

I spent about 45 minutes trolling the internet - in particular, the USDA and CDC websites - trying to come up with count data for how many animals are tested for rabies in Wyoming in a given year. My searches took me here and here. I found data from 2010. 32 wild animals tested for rabies in Wyoming in 2010 - 2 were cattle, 12 were bats, and 20 were skunks. I could not locate more recent data or 2010 data on the total number of animals tested, which posed a problem, because we are really interested in the total number of animals tested to inform us on what proportion of rabies cases we expect. I wanted to reverse engineer the actual percentage of rabid skunks we expect in Wyoming... how high does the percentage of positive tests have to be before we release such a warning to the public? 50%? 60%? 75%?

[side note: I learned a great deal about how rabies spreads and which animals are typically affected. See the map below.]









U.S. map showing which animal is the most frequent rabies carrier by region [source: CDC]

I asked the students to use their TI-84 calculator and, through trial and error, to determine the population proportion value p (the actual proportion of rabid skunks) that would lead to NOT rejecting the null hypothesis for each alpha level. Here's a slide with the work a student wrote on the board:



I had students use trial and error, with the TI-84 one-proportion z-test, to determine what the decision would be (whether we would reject or fail to reject the null hypothesis) for different values of the assumed population proportion p.

For example, with an alpha level of .05 and assuming the population proportion of all skunks that have rabies is 50%, we would reject the null - the population proportion being 50% and conclude in favor of the alternative, that the proportion of skunks with rabies is likely higher than 50% based on the sample, since we expect to see a sample this extreme (16 out of 19 skunks rabid) only 14 times out of every 10,000 samples due to chance alone.

Through trial and error, my students found we would not reject the null hypothesis given a sample of 16 out of 19 skunks being rabid if the actual proportion of rabid skunks was 66.37% or higher. 2/3 of all skunks being rabid is pretty scary to think about, but if we are trying to determine why the USDA released a rabies warning, this is useful information for us.

I had students formulate the idea of a Type I error (mistakenly rejecting a true null hypothesis) and a Type II error (mistakenly failing to reject a false null hypothesis) in the context of this problem through discussion.

Type I error would mean the proportion of skunks with rabies is greater than the assumed proportion when in fact the assumed proportion is true. This would mean potentially raising a false alarm, or releasing an outbreak alert when one is not needed.

Type II error would mean concluding the proportion of skunks with rabies is the assumed proportion when in fact the actual proportion of skunks with rabies is higher. This would mean not alerting the public to a potential outbreak when in fact an outbreak is actually going on.

Students collectively agreed Type II error is more serious in this setting. The students discovered through this activity the decision they should make about the alpha level - maximizing the chance of a Type I error in favor of decreasing the chance of a Type II error.

My students breezed through the exam covering Type I and Type II error for the first time in my career. 🙂

I have been in Washington. DC for the past few days at the Teaching & Learning 2014 conference in our nation's Capitol.

When I stood at this podium, our meeting and session with policy makers had just ended. No audience occupied the opposite side of the podium. I began thinking, however, about the opportunity for positive change an educator could bring to a public office. Towards the end of the conference, we had the privilege to hear Angela McLean. Here is an article about Angela.


Below are some of the resources and knowledge from the conference sessions I attended.

Ralph Smith's Grade Level Reading Campaign



Myself, Deborah Ball, and colleague Dan Schaben after Deborah's outstanding presentation.

Deborah Ball on "Safe to Practice"

Linda Darling-Hammond's Getting Teacher Evaluation Right


Sarah Brown Wessling, featured on the site Teaching Channel, helping teachers connect with great instructional examples


One thing I hope my students learn in school is how a person can unlock their creative potential and pursue their passions into adulthood. To this end, our Math Theory & Problem Solving (MTPS) class took a field trip to do some data collection. I would like to give special thanks to Daryl Payne for allowing our MTPS class to enjoy data collection (racing cars). Daryl's creativity and passion for racing inspired the students to re-imagine what is possible in the world outside school. The video below shows the electric car race track where students raced.

We spent our lunch period having a pizza party prior to racing cars as a reward for the work students have been doing in class. My original motivation in this trip was collecting data and trying to determine how a person could use statistics to potentially detect cheating through exceptional lap times. However, there are also many other mathematical and statistical ideas we can explore with what we learned on this trip.

Here are some things that students wondered about and could lead to mathematical investigations:

  • How much longer is the outside lane than the inside lane?
  • How does the electronic timing system work?
  • How much voltage/current is being supplied to each car?
  • What would a person have to know about electronics and circuitry in order to build such a track?
  • What is the difference between cars with magnets and cars without? (Cars with magnets can maintain higher speeds around the turns, for example)
  • What amount of voltage causes a car to fly off the track?
  • How should we determine the best racer? The fastest lap time? The best median lap time?
  • Is there a difference between the performance of the blue guest car and the silver guest car? If so, how could we detect this difference numerically?
  • How does the "KILL POWER" switch work?


Below, Mr. Payne gives the students some guidelines to follow while they practice racing on the race track.

This short video shows the beginning of a head-to-head race between students.


I am interested to see the types of mathematical investigations that spring up from our field trip. We will take our race data and use it to determine the best racers.