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 5.75 6.00 6.75 7.25 7.50
y 61 41 62 81 98
r=0.855
x-bar=6.65
~y68.6
y=-93.145+24-323x
what percentage of the variation in y cannot be explained by the corresponding variation in x and the least-squares line?
(Points: 4)
2.1%
7.2%
26.9%
73.1%
85.5%
You wish to find the coefficient of nondetermination, which is the percentage of the variation in y that cannot be explained by the corresponding variation in x.
Formula: 1 - r^2
I'll let you take it from here. Convert the decimal to a percentage for your answer.
0.945
To determine the percentage of variation in y that cannot be explained by the corresponding variation in x and the least-squares line, we need to calculate the coefficient of determination (R^2).
R^2 represents the proportion of the variation in the dependent variable (y) that can be explained by the independent variable (x) and the least-squares line. It ranges from 0 to 1, where 0 indicates no relationship and 1 indicates a perfect relationship.
To find the percentage of variation that cannot be explained, we subtract R^2 from 1 and multiply by 100.
Given that the correlation coefficient (r) is 0.855, we can calculate R^2 as R^2 = r^2.
R^2 = (0.855)^2 = 0.731
Therefore, the percentage of variation in y that cannot be explained by the corresponding variation in x and the least-squares line is:
(1 - 0.731) * 100 = 0.269 * 100 = 26.9%
So the correct answer is 26.9%.
To find the percentage of the variation in y that cannot be explained by the corresponding variation in x and the least-squares line, you need to calculate the coefficient of determination, also known as R-squared.
R-squared is the square of the correlation coefficient (r), which measures the strength and direction of a linear relationship between two variables. In this case, r is given as 0.855.
So, to find R-squared, you square the correlation coefficient:
R^2 = r^2 = 0.855^2 = 0.731
R-squared represents the proportion of the variance in y that can be explained by the linear regression model (the least-squares line). Therefore, the remaining proportion (1 - R^2) indicates the percentage of the variation in y that cannot be explained.
1 - R^2 = 1 - 0.731 = 0.269
Converting 0.269 to a percentage, we get 26.9%.
Therefore, the correct answer is:
26.9%