Question

In: Math

It is thought that prehistoric Indians did not take their best tools, pottery, and household items...

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 5.50 6.25 6.75 7.25 7.50 y 9 38 38 50 72 What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line?

Select one:

a. 89.9%

b. 10.1%

c. 94.8%

d. 0.3%

e. 1.0%

Solutions

Expert Solution

X Y (x-x̅)² (y-ȳ)² (x-x̅)(y-ȳ)
5.5 9 1.3225 1049.76 37.26
6.25 38 0.16 11.56 1.36
6.75 38 0.01 11.56 -0.34
7.25 50 0.36 73.96 5.16
7.5 72 0.7225 936.36 26.01
ΣX ΣY Σ(x-x̅)² Σ(y-ȳ)² Σ(x-x̅)(y-ȳ)
total sum 33.25 207 2.575 2083.2 69.45
mean 6.65 41.4 SSxx SSyy SSxy

sample size ,   n =   5      
here, x̅ =   6.65       ȳ =   41.4
              
SSxx =    Σ(x-x̅)² =    2.575      
SSxy=   Σ(x-x̅)(y-ȳ) =   69.45      
R² =    (Sxy)²/(Sx.Sy) =    0.8992

so, 89.92 %variation in y can be explained by the corresponding variation in x and the least-squares line

option a) is answer


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