Question

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?

Select one:

a. 10.8%

b. 4.2%

c. 89.2%

d. 20.4%

Answer 1

The statistical software output for this problem is:

Simple linear regression results:
Dependent Variable: y
Independent Variable: x
y = -64.818898 + 16.031496 x
Sample size: 5
R (correlation coefficient) = 0.89215768
R-sq = 0.79594533
Estimate of error standard deviation: 8.3505858

Parameter estimates:

Parameter	Estimate	Std. Err.	Alternative	DF	T-Stat	P-value
Intercept	-64.818898	32.784304	≠ 0	3	-1.977132	0.1425
Slope	16.031496	4.6864618	≠ 0	3	3.4208102	0.0418

Analysis of variance table for regression model:

Source	DF	SS	MS	F-stat	P-value
Model	1	816.00315	816.00315	11.701942	0.0418
Error	3	209.19685	69.732283
Total	4	1025.2

Hence,

Precentage of variation that cannot be explained

= 100% - R-square

= 100% - 79.6%

= 20.4%

Option D is correct.