Rating Movies on a Numerical Scale: My wife and I often go to movies and afterwards assign


Rating Movies on a Numerical Scale: My wife and I often go to movies and afterwards assign a rating ranging from0 to 10 to the movie we saw.

A: Suppose we go to see a double feature, first “Terminator 2”with the great actor Arnold Schwarzenegger and then the adaptation of Jane Austin’s boring novel “Emma”. Afterwards, you hear me say that I rated “Terminator 2” as an 8 and “Emma” as a 2, and you hear my wife comment that she rated “Terminator 2” a 5 and “Emma” a 4.

(a) Do my wife and I agree on which movie is better?

(b) How would your answer change if my wife’s ratings had been reversed?

(c) Can you tell for sure whether I liked “Terminator 2”more than my wife did?

(d) Often, my wife and I then argue about our rankings. True or False: It makes little sense for us to argue if we both rank one movie higher than the other even if we assign very different numbers.

B: Suppose that the only thing I really care about in evaluating movies is the fraction of “action” time (as opposed to thoughtful conversation) and let the fraction of screen time devoted to action be denoted x1. Suppose that the only thing my wife cares about when evaluating movies is the fraction of time strong women appear on screen, and let that fraction be denoted x2. “Terminator 2” has x1 = 0.8 and x2 = 0.5 while Emma has x1 = 0.2 and x2 = 0.4.

(a) Consider the functions u(x1) = 10x1 and v(x2) = 10x2 and suppose that I use the function u to determine my movie rating and my wife uses the function v. What ratings do we give to the two movies?

(b) One day I decide that I will assign ratings differently, using the function u(x1) = 5.25x1/6 1 . Will I rank any pair of movies differently using this function rather than my previous function u?

What approximate values do I now assign to “Terminator 2” and “Emma”?

(c) My wife also decides to change her way of assigning ratings to movies. She will now use the function v(x2) = 590x26.2. Will her rankings of any two movies change as a result? What approximate values does she now assign to the two movies?

(d) Suppose my wife had instead chosen the function v(x2) = 10(1−x2). Will she now-rank movies differently?

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