IPS - Day 41

Today we continued working on the "What Do Students Drive?" investigation. To start things off, I had students finish putting together histograms to present to the class. Several students shared their graphs. I used this as an opportunity to provide commentary on what constitutes a properly constructed graph.

I was pleased that students remembered to keep their bin sizes consistent. As for scale, many students simply used the count as units for the x-axis. I had to point out that we were interested in looking at the percentage of cars manufactured by region and therefore the x-axis scale should be consistent with the item of interest, in this case a percent.

Once we viewed these graphs, I explained how a histogram could be rendered using a graphing calculator. I have TI Smartview software installed, so I am able to demonstrate this process for TI-83 and TI-84 family of calculators, which most students in class possess.

Besides showing students how to construct a histogram on the calculator, I used this time to discuss different bin widths. For this class, I demonstrate how to vary bin sizes using the x-scale in the window setting. I change the bin size and then display the graph. I showed that as the bin size narrows, the graph provides more detail but looks more jumbled. I then show that as the bin size increases the graph becomes more bunched and loses detail. The objective is to find a bin size that provides enough detail while still producing a graph that is explainable, hopefully a graph that is unimodal and somewhat symmetric.

Next, I reviewed where we were in the analysis process. I emphasized that the histograms that the class created showed what could happen randomly under the conditions of their assumed hypothesis. In other words, the histograms present a conditional probability distribution for random events under the hypothesized percentages. We can use this as a basis for comparison. The histograms describe what random events should look like if our hypothesis is true. We collect a random sample and compare its values to the distribution we generated. We can even calculate the probability of seeing a sample value as extreme as the one we are seeing.

Students then worked in their groups to create a sampling plan. I have a schematic of the student parking lot and provide this as an aide in the development of their plan. I must sign off on their plan before they may proceed to collect data. I instructed students to develop a plan that would generate a random sample since we need to compare the collected random sample to what we believe random samples should look like.

Several groups planned on using a systematic sample. When I asked them how the chose their starting spot, they invariably replied that the just picked it from convenience. I explained to them this was not random. Students then asked if the randomly picked a starting point if they could then proceed systematically from there? I responded they could as long as they continued with this process through the entire list until they cycled back to where they started.

Other groups decided to use a simple random sample and generated random values within the proper range. One student used his calculator to generate the list and then sort it. This allowed the group to work through their list in a sequential order and saved them time running all over the parking lot.

One group split the parking lot in half and just wanted to select cars from within one half. I told them this was not random. They didn't understand why not. I explained that we didn't know how students were assigned spots. Perhaps all of the seniors were assigned spots in the half they were using and juniors were assigned spots in the other half. Or, maybe the spots were assigned by car type, so that only foreign made cars were assigned to the half they wanted to use. The purpose of having random selection is to remove unknown factors that may inadvertently affect the sample and introduce bias. This group better understood the importance of randomness in the sample.

A last group really struggled with the ideas of random sampling. Finally, one member asked if they couldn't just select the cars from one row. I asked how they would determine which row to use. After some thought the student said that all the rows could be numbered and that one of the rows could be randomly selected. I pointed out that this was a cluster sample.

Students went out and collected the make and color of cars in the student parking lot. Thankfully the weather was pleasant and students were anxious to collect their data. Once students returned, I presented the next steps in their analysis and the requirements for comparing their sample results to their hypothesized distribution.

Next class, students will conduct their analysis and develop a written report for their results.

Visit the class summary for a student's perspective and to view the lesson slides.