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	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. 0.3%

b. 89.9%

c. 1.0%

d. 10.1%

e. 94.8%

Answer 1

The best linear regression line is y = -137.96 +  26.97 * x.

and 89.9% percentage of the variation in y can be explained by the corresponding variation in x.

x <- c(5.50,6.25,6.75,7.25,7.50)
> y <- c(9,38,38,50,72)
> fit <- lm(y~x)
> fit

Coefficients:
(Intercept) x
-137.96 26.97

> summary(fit)

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -137.956 34.880 -3.955 0.0288 *
x 26.971 5.215 5.172 0.0140 *

Residual standard error: 8.368 on 3 degrees of freedom
Multiple R-squared: 0.8992,   Adjusted R-squared: 0.8655
F-statistic: 26.75 on 1 and 3 DF, p-value: 0.01403