QR 26: CHOICE AND CHANCE

Assignment #4: Scales and Quantification

1. Study the UMAP module on "Measurement Scales" by Growney. Although we will not spend time on it in class, look through Chapter 6 of Making Hard Decisions.
2. Keep working with Excel and PrecisionTree or TreePlan so that you feel comfortable doing basic calculations and trees using these tools, including the Dumond case below.
3. Problems from the Growney hand out to hand in: #2, #3, #8.

GPA: Serving on national fellowship selection committee, you notice that the system for scaling letter grades to compute class averages is different at Harvard from other colleges. If student record x ranks higher than y at Harvard, can you conclude that the same grade record x would also rank higher than y at other colleges? Do it make sense to say that Record x is 10% better than record y? That Record x is twice as strong as Record y? Does it make a difference if the two averages are computed on the same or different scales? (Most schools other than Harvard assign 4.0 to an A, 3.7 to an A-, 3.3 to a B+, etc. Using Excel, try considering under both systems a record x with one A, 12 A-, 12 B-, and 7 C- grades verses record y with 19 A, 1 B and 12 C+ grades.) Explain what is going on here. What would you propose to do about this? How would you present and justify your proposal?

Temperature: Starting from the assumption that both are interval scales, derive formulae for switching between Fahrenheit and Centigrade by imposing what you know about the freezing and boiling points of water in each system. Show that, in contrast to the previous example, the question of whether the average temperature (as measured each day at noon, say) in City X is bigger than in City Y over a given period does not depend on which scale you use in both.

Preferences: You are decorating a room for an indecisive grandparent, who will not tell you what color paint to use, but will give you a strict and consistent preference between any pair of colors. The only choices are Lilac, Aqua, Brown, Mushroom, White, Green, and Pink. Using first letters as abbreviations, you have asked seven questions and learned that your grandparent holds: M>P, M>G, P>W, A>M, G>W, B>L, and L>M. What is the next and last question you ask in order to determine what color to paint the room? In all, you have asked eight questions. What is the least number you could have asked, and is there a strategy for questioning that ensures you only ask the least number possible? Justify your answer.

D. Prepare notes you can discuss and hand in for the Dumond case on pp. 211-213 of MHD.

Read at least chapter 1 of Theodore Porter's Trust in Numbers, and enough of a book such as Innumeracy by John Allen Paulos to debate about the role of numbers in modern society. For example, is numeracy essential to all citizens as Paulos argues, or just an administrative tool for appearing objective as Porter seems to believe?

E. Think about precisely what you find most and least compelling about using decision trees.