In IB Mathematical Studies, students have to be able to complete a Chi Squared Test to determine independence (or not) of two variables.
The full process of carrying out a Chi Squared test is quite long, and so I decided to break it down into small steps, and get students to master each step before adding on the next one. The steps are listed in the image below, which I showed to students at the start of the unit.
This year, with my focus on example-problem pairs, I decided to get students to use 8 separate pages for each of the questions (1 example, 1 your turn, and 6 from an exercise). Each page would become a full Chi Squared test, but we would only fill in one step at a time.
I showed students the image below as a rough guide to how each page would look, so they could get an idea of what I meant.
Then, for each stage I went through a bit of background, and then showed the example through my visualiser on a sheet of paper (the completed example is shown below).
In the first lesson I started by introducing the idea of hypotheses for a statistical test, and more specifically that the null hypothesis will assume that two variables are not related.
After going through the example, students did a your turn, which we checked together, before they proceeded with the exercise of 6 questions. They cut out each question and stuck it at the top of a page.
Next we looked at the idea of a contingency table, and the need for finding the totals for each row and column. I decided not to look at putting raw data into a table at this stage, and will cover this at a later stage, in preparation for their internal assessment project.
I then gave out the second worksheet which included the tables for the six questions above, and they stuck out the tables and stuck them below the hypotheses they wrote in step 1.
Next we looked at the concept of degrees of freedom as the number of values which are free to vary (those in red) as opposed to those which are fixed by the chosen values (those in green). After showing this for the example, students filled in step 3 for the your turn, followed by the 6 questions.
Next we calculated the expected frequencies for the contingency table. After a brief discussion about the probability (this is a later unit, so I did not stress it here), we worked out the expected value for each cell in our tables. We noted that in the example, because the number of Males and Females was equal, the expected frequencies of each were equal.
At the start of the next lesson we recapped the process so far, with students working in pairs on the whiteboards around the room.
Step 5 is calculating the chi-squared value for the data, and we went through how to do this by hand. I did the calculation for the first cell, then asked pairs to do the other cells and check each other. Finally, with all the values, we summed them up to get a value. I really emphasised here that the chi squared value represents the total error of the observed values from what we expect (if the null hypothesis is true). This is something I have not really focused on in the past, but felt that giving the abstract chi squared value a concrete purpose (to measure error) helped students understand what it represented.
After this we went through the process on our graphical calculator to see that in reality we can find the value without needing to go through this whole process every time.
Before moving on to step 6, we discussed that we want the error to be small in order to agree with the null hypothesis, and more importantly, if the error is big, then we can say that the null hypothesis is not true. At this stage several students asked how big is too big an error, and I told them that we will be seeing how we decide this in the next couple of steps. This was definitely the right decision at the time, as I did not want to get bogged down in the details of this before they could work it out, and retrospectively, it worked really well.
Using the critical value to determine the conclusion of the test was next, and the reference to large errors being not what we want paid off here, as students were happy with the fact that we reject the null hypothesis if it is greater than the critical value
Step 7 looked at p-values, and again, I placed a big emphasis on the fact that this is the probability of it happening if the null hypothesis is true. So a small probability means it is likely the null hypothesis is false.
We now briefly discussed how we can find the critical value using the degrees of freedom and the significance level we are interested in, and how this told us how big an error was too big, depending on how confident we wanted to be in our answer.
At this point I started using this diagram to compare the two ways of concluding whether to reject or not the null hypothesis.
Finally, we looked at interpreting our conclusions in terms of the problem, linking back to the hypotheses stated.
As this process went over several lessons, there was lots of time to retrieve previous steps of the process, and by the time we got to making the conclusions, all students were able to perform all the previous steps. I made use of my whiteboards in these review questions, projecting a question going up to the step we had reached, and getting pairs to work it out on the whiteboards. This allows me to quickly and easily see any misconceptions, and for students to check against the other groups.
To finish the unit, I pieced the whole process together, and set some exam questions for the students to do.
The benefits of breaking down this long process are:
- Each individual step was very accessible
- Time to retrieve earlier steps was built in
- Students could experience a lot of success very quickly
The downsides were:
- It took longer to get through the whole process
- Students lost the aim of the process due to this length of time
I was able to address the second of these two downsides when bringing the whole process together, and talk more about why we use Chi squared tests, and what they tell us.
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