28. It is thought that prehistoric Indians did not take their best tools, potter

Question

28.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) for an archaeological site in the southwestern United States. Let y be the percentage of unidentified artifacts (no specific cultural affiliation) at a given elevation. Suppose that the following data were obtained for a collection of archaeological sites in New Mexico:

x

6.00

6.25

7.00

7.25

8.25

y

33

38

35

58

69

What percentage of the variation in y cannot be explained by the corresponding variation in x and the least-squares line?

x

6.00

6.25

7.00

7.25

8.25

y

33

38

35

58

69

Explanation / Answer

The table to calculate the value of coefficient of determination.

First we will calculate correlatio coefficient

r = [ n(xy) - (x)((y)]/ sqrt [ n (x2 ) - (x)2 ]*[ n (y2 ) - (y)2 ]

r = [ 5 * 1670.25 - 34.75 * 233] / sqrt [ (5 * 244.6875 - 34.752 ) * ( 5 * 11883 - 2332)]

r = 254.5/ sqrt (15.875 * 5126)]

r = 254.5/ 285.2635

r = 0.8922

Coefficient of Determination R2 = 0.796

so, 79.59 % percentage of the variation in y cannot be explained by the corresponding variation in x.

The least - squares line y^ = a + bx

a = [(y) (x2 ) - (x) (xy)]/ [ n (x2 ) - (x)2 ]

a = (233 * 244.6875 - 34.75 * 1670.25) / (5 * 244.6875 - 34.752)

a = -1029/ 15.875 = -64.82

b =  [ n(xy) - (x)((y)]/ [ n (x2 ) - (x)2 ]

b = [5 * 1670.25 - 34.75 * 233] / (5 * 244.6875 - 34.752)  

b = 254.5/ 15.875 = 16.0315

y^ = 105.2125 + 16.0315x

x y x^2 y^2 xy 6 33 36 1089 198 6.25 38 39.0625 1444 237.5 7 35 49 1225 245 7.25 58 52.5625 3364 420.5 8.25 69 68.0625 4761 569.25 sum 34.75 233 244.6875 11883 1670.25

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