2. Opfer and Segler (2oo7) asked second- and fourth-grade school children to mark on a number line where a given number would fall. Each child was given a drawing ofa number line with two ends marked at o and 1ooo, and was then asked to make an X on that line where a number, for example 15o, should be placed. They asked each child to place several different numbers on the number lines, each on a fresh new piece of paper. 0 1 000 ISO The researchers then measured the placement of each mark on a linear scale. The results, averaged over all 93 kids for each group, are given in the le "numberline.csv". a. Plot the fourth graders' guesses against the true value. Is this relationship linear? If not, find a transformation of X or Y that converts the relationship into an approximately linear one. b. Plot the second-graders guesses against the true value. Is this relationship linear? If not, find a transformation of X or Y that converts the relationship into an approximately linear one. Fit a linear regression to both the transformed and untransformed data. Examine the residual plots for both the transformed and untransformed data. c. Assume that the difference between the shapes of these curves is real. What would you conclude about the difference in the way 2\"\"| graders and 4th graders perceive numbers