At this point we should not have to tell you in every problem that you need to explain your answer. We are hoping that writing it here, as a general reminder, will be good enough. These problems will help you prepare for the test.

1. A news story reported recently that at the zoo in Racine, WI, a lioness gave birth to four lion cubs. The caption on a photo of the cubs said:

   Transvaal lions are native to South Africa, and it is unusual for a lioness to have four cubs. The average litter is three..

   1. Which average (mean, median or mode) do you think the newspaper is using? Why do you think so? Does it make sense that a litter of four cubs would be unusual? Explain.

   2. We were curious about this statistic and asked Google to search for

      	lion litter size
      

      We rapidly found the paper The Lack Clutch in a Communal Breeder: Lion Litter Size is a Mixed Evolutionarily Stable Strategy in The American Naturalist, Vol. 145, No. 5. (May, 1995), pp. 833-841. You don't have to read it, but if you want to we have put a copy at this link.

      The paper contains this graph:

      

      Notice that the vertical axis is labeled "frequency" but the values are numbers between 0 and 1. Each of those numbers represents the probability that a litter will be a particular size.

      We can estimate the proportion of litters studied for the Serengeti Woodlands lions, using the graph above.  The data point for litters of size 1 is at height about 0.17, which we have entered in the table below.

          Litter Size    proportion of litters studied (Serengeti Woodlands lions)
               1                0.17
               2
               3
               4
               5
               6
      
      

      Complete this as a table in Excel.

      Since the legend on the graph tells us how many litters in each location were counted (the value of N) we can figure out the actual frequencies from the probabilities. For example, 199 litters in the Serengeti Woodlands were studied. so there were 199*0.17 ~ 34 litters of that size. So in the following table we can fill in line 1:

          Litter Size    Number of litters studied
               1                34
               2
               3
               4
               5
               6
      
      

      Make Excel complete that table for you by putting the appropriate formula in the upper left hand corner (where there's a 34) and copying that formula to all the other cells in the table.

      Create a column graph of the data. Label your graph as usual. Print the spreadsheet and turn it in with this homework.

   3. Does the shape of your graph match the shape of the graph of the Serengeti Woodlands lions in the American Naturalist paper? Should it?

   4. Why does a column graph like the one you made better illustrate the data than the line graph in the paper?

   5. To return to the original question that started us down this path: using all of this information, do you think that it is unusual for a lioness to have a litter of four cubs?

2. Read Chapter 2 (The Well-Chosen Average)in How to Lie with Statistics. In this chapter, the author describes a fictional manufacturing business that has paid out $198,000 to the ninety employees who work there. Read through this section carefully (and study the graphs).

   1. To convince us that you understand this section, show us how the following statistics were calculated:

                  	Average wage of employees......  $  2,200
                  	Average salary and profit of owners..  $ 26,000
      
                  	Average wage or salary.........$  2,806.45
                  	Average profit of owners........$ 5,000
      

   2. Why would the owners of the company use the second set of figures rather than the first set? Are they lying when they do this?