Day 67: Speed Dating Random Variables
Today in AP Stats we tackled combining random variables. This has always been a “dry” topic, even when I’ve had the kids complete an activity….every year it seems the “big idea” gets lost on them in the minutia of collecting data by hand and crunching the numbers. So I used this example from our teacher resources, but used an idea I saw Josh Tabor use in his class notes of using a simulation to lay the foundation for developing the concepts and resultant formulas. It proved to be a great way to review simulation AND introduce the RandNorm( function on the Nspires.
First we talked through developing the simulation. Darn if the kids are still struggling with quantifying a situation…food for thought for later. But we finally set up the parameters (and set them up for two sample inference to boot! See the probability statements I stuck in)
Secondly, we used the power of the Nspires to collect 1000 random samples for the males and the females. I had the students verify that their set of numbers (at least the first few) were not the same as their groups. Then we used the probability statement to determine the value of the combined random variable M – F. We looked at the values in the table first, noticing that there were positive and negative differences. I asked if it looked about even between positive and negative values and they said they couldn’t tell…yay! So we graphed them and the collective “it looks like a normal curve” was exclaimed…yay again! We also used our probability statement, P(M-F>0) to draw in the vertical line and again estimate the probability. We sorted the “d” list and then counted the number of negative values…ugh, no fun…and then determined the ratio and then the complement. So much good stuff!
Thirdly, we calculated the means of the three random variables and looked for the (obvious) relationship, which they found in a heart beat. Asked them if this was the true population values and they chimed in “NO” only an approximation since this is a sample…helped that everyone had slightly different values but all close to the true values of μM = 68.5 and and μF = 64. They also were able to reason through how to find the value of μM-F = 4.5.
The standard deviation was a little different story. Again they saw that their values were close to the true values of σM = 4 and σF = 3, but σM-f = 5 not 1. So we had a chat about what happens to the variablility when we subtract variables; i.e. that subtracting does not reduce variability but actually increases it. Then a student blurted out “pythagorean theorem!” So we explored that idea and developed a generalization as shown below. All in all, a great lesson and I believe my “statisticians” turned “mathematicians” will remember this concept.