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	6.50	7.25	8.00	8.75
y	10	41	53	89	89

What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line?

Group of answer choices

2.6%

0.3%

94.8%

0.1%

97.4%

Answer 1

Here in this scenario 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.

Now based on given data we need following table to calculate the Coefficient of determination and least squares equation.

The independent variable is XX, and the dependent variable is YY. In order to compute the regression coefficients, the following table needs to be used:

	X	Y	X*Y	X2	Y2
	5.50	10	55	30.25	100
	6.50	41	266.5	42.25	1681
	7.25	53	384.25	52.5625	2809
	8.00	89	712	64	7921
	8.75	89	778.75	76.5625	7921
Sum =	36	282	2196.5	265.625	20432

Based on the above table, the following is calculated:

To calculate the coefficients of determination first we need to compute the correlation coefficient between x and y.

The correlation coefficient is calculated as below,

The correlation coefficient between x and y is 0.974. Which is strong positive correlation coefficient between x and y Variable.

Now the coefficient of determination is simply square term of correlation coefficient.

Coefficient of determination = R^2 = (0.974)^2

R^2 = 0.9486.

percentage of the variation in y can be explained by the corresponding variation in x is

The correct answer choices is

C) 94.8%

There is 94.8% of variation in y Variable explained by x independent variable.

Now the least squares regression equation is calculated using following steps,

the least squares regression equation is Y = -129.7354 + 25.8521 X.

Thank you.