The other day, you went online to look at how other cities around the world compare to San Diego in terms of the climate. You looked up Osaka, Japan and found that their temperature is recorded in Celsius. The table below shows the monthly temperatures in these two places.

SanDiego,US(F) 66 66 67 69 69 72 76 77 77 74 70 66 Osaka,Japan(C) 9 10 14 20 25 28 32 33 29 23 18 12

You want to compare the median monthly temperatures in San Diego and Osaka. One way is to convert all the degrees in Fahrenheit to Celsius before computing the median temperature for San Diego. This way, you compare the median temperatures in both places using the same units.

You friend Skip is lazier and thinks you can skip some of that work: "Why don't we find the median temperature in San Diego in Fahrenheit first, and then only convert the median value to Celsius? That way we only need to do the conversion once or twice."

a) b)

Is Skip correct that you'll get the same median temperature either way? Show your work.

More generally, letg(t) be the function which takes in a temperature in degrees Fahrenheit and outputs the temperature in Celsius. That is,g(t) =95(t32). Then this problem can be formulated mathematically as the validity of the following equation:

Median(g(t1), . . . , g(tn)) =g(Median(t1, . . . , tn)).1

Prove this equation or disprove it by counter-example.

c)Finally, you find that the median temperatures for San Diego and Osaka are quite close to each other. Can you say that the climate at the two cities are similar to each other? Where do you think the median temperature is more representative of the overall climate, San Diego or Osaka? What is your reasoning?