The paper “Developmental and Individual Differences in Pure Numerical Estimation” (Developmental Psychology [2006]: 189–201) describes a study of how young children develop the ability to estimate lengths. Children were shown a piece of paper with two lines. One line was a short line labeled as having length 1 zip. The second line was a much longer line labeled as having length 1000 zips. The child was then asked to draw a line that had a length of a specified number of zips, such as 438 zips. The data in the accompanying table gives the length requested and the average of the actual lengths of the lines drawn by 30 second graders.

Requested Length Second Grade Average Length Drawn 3 37.15 7 92.88 19 207.43 52 272.45 103 458.20 158 442.72 240 371.52 297 467.49 346 487.62 391 530.96 438 482.97 475 544.89 502 515.48 586 595.98 613 575.85 690 605.26 721 637.77 760 674.92 835 701.24 874 662.54 907 758.51 962 749.23

a. Construct a scatterplot of y second grade average length drawn versus x requested length.

b. Based on the scatterplot in Part (a), would you suggest using a line, a quadratic curve, or a cubic curve to describe the relationship between x and y? Explain your choice.

c. Using a statistical software package or a graphing calculator, fit a cubic curve to this data and use it to predict average length drawn for a requested length of 500 zips.

y^ 5 138.471 1 19276x 2 0.0032x2 1 0.000002x y^ 5 555.219