Data for this problem are based on information taken from Prehistoric New Mexico: Background for Survey (by D. E. Stuart and R. P. Gauthier, University of New Mexico Press). It is thought that prehistoric Indians did not take their best tools, pottery, and household items when they visited higher elevations for their summer camps. It is hypothesized that archaeological sites tend to lose their cultural identity and specific cultural affiliation as the elevation of the site increases. Let x be the elevation (in thousands of feet) of an archaeological site in the southwestern United States. Let y be the percentage of unidentified artifacts (no specific cultural affiliation) at a given elevation. The following data were obtained for a collection of archaeological sites in New Mexico:

Complete parts (a) through (e), given ˆ‘x = 31.25, ˆ‘y = 164, ˆ‘x2 = 197.813, ˆ‘y2 = 6832, ˆ‘xy = 1080, and r ‰ˆ 0.913.
(a) Draw a scatter diagram displaying the data.
(b) Verify the given sums ˆ‘x, ˆ‘y, ˆ‘x2, ˆ‘y2, and ˆ‘xy and the value of the sample correlation coefficient r.
(c) Find x, y, a, and b. Then find the equation of the least-squares line ŷ = α + bx.

(e) Find the value of the coefficient of determination r2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained?
(f) At an archaeological site with elevation x = 6.5 (thousand feet), what does the least-squares equation forecast for y = percentage of culturally unidentified artifacts?

Transcribed Image Text:

x 5.25 5.75 6.25 6.757.25 y 19 3 333762