Studies have shown that certain ants in a colony are assigned foraging duties, which require them to come and go from the colony on a regular basis. Furthermore, if y is the colony size and x is the number of ants that forage, the relationship between y and x has the form y = ax^b, where a and b vary from species to species. Once the parameter values have been estimated for a particular kind of ant, biologists can count the (relatively small) number of ants that forage and then use the regression equation to estimate the (much larger) number of ants living in the colony. The table on p. 554 gives the results of a “calibration” study done on the red wood ant (Formica polyctena): Listed are the actual colony sizes, y, and the foraging sizes, x, recorded for fifteen of their colonies (94).

(a) Find a and b using the sums below.

(b) If the number of red wood ants seen foraging is 2500, what would be a reasonable estimate for the size of the colony from which they came?

Foraging Size, x                                 Colony Size, y

45………………………………         280

74………………………………         222

118……………………………..         288

70………………………………         601

220……………………………..         1,205

823……………………………..         2,769

647……………………………..         2,828

446……………………………..         3,229

765……………………………..         3,762

338……………………………..         7,551

611……………………………..         8,834

4,119……………………………       12,584

850……………………………..        12,605

11,600…………………………..       34,661

64,512…………………………..       139,043

15 15 10g Xi-41.77441 log yi 52.79857 (log= 126.60450 1og xi . logy;= 156.03811 i-l