Chapter 7: A Tale of Two Variables, Continued (Day 54)

Today in Stats, I begin to prepare the class for the quiz tomorrow. This quiz will cover Chapter 7 of the text, on bivariate data, scatterplots, and correlation coefficients r. First of all, I started by having the students calculate a value of r for a set of data -- with no calculator! The process isn't too difficult, but it's definitely time-consuming. Here are the data we used for this calculation:

(1,10), (2,4), (2,10), (3,10), (4,20), (7,20), (7,26), (12,28), (12,30), (12,32), (15,30)

These aren't random values -- instead, they were selected so that the means (7 for the x-values, 20 for the y-values) and standard deviations (5 for the x-values, 10 for the y-values) are whole numbers. The most difficult calculation is to find 16 * 16 = 256, which I give them in advance. (Some teachers do teach the perfect squares up to at least 16^2). Thus the only numbers that we're required to divide by are 5, 10, and 11 (the number of data points -- notice that n - 1 is 10, which is an easy divisor). Many of the z-values are 0, +1, or -1 -- and in fact, every product of z-values includes 0, +1, or -1. The final value of r is 0.92, indicating a strong positive correlation. The whole calculation took us about 45 minutes.

Why did I have the students make this calculation -- after all, r coefficients are almost never calculated by hand without a calculator? It's mainly to get the students to practice their arithmetic skills -- adding, subtracting, multiplying, and dividing without a calculator.

Earlier this week, I also chose random data in order to have the students estimate r coefficients. Here I ostensibly chose random points between (0,0) and (12,12) -- but in reality, I didn't. Instead, I chose points with both x and y less than 6, and then both coordinates more than 6. Believe it or not, choosing random points in these two quadrants produced an r value exceeding 0.8. And dividing the graph into nine regions, and going from the upper left to lower right (in order to force a negative r), led to r almost exactly -0.9. 

Then again, recall that the definition of r is based on quadrants, so this shouldn't be surprising. Also, by calculating an r value by hand, I can see how the value of r approaches 1 (or -1) if the points are collinear.

After finding r, I had the students discuss even questions from the text -- and this leads to problems during our discussion. The first two questions, #2 and #4, ask subjective questions such as, if we are plotting number sold vs. price, which goes on the x-axis and which on the y-axis? It depends -- it makes sense to plot price on x as we are given a price and can ask how many was sold at that price (and this is what the answer key states), but then again, a demand curve in Economics would place price on y. And as #2-4 have four parts each, that's eight subjective questions to discuss -- taking time away from #6-10, which are scatterplot questions that will appear on the quiz. I end up dropping #4 from the assignment, just to make sure that we get to #6.

The review in Calculus goes a little more smoothly. There are four DeltaMath assignments this week -- first I call on students to help me do some problems on the board, and then I let them loose, giving them time to do some of the work on their own. This helps me decide which questions to include on the quiz.

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