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: $$\begin{array} { c | c c c c c }
\hline x & 5.75 & 6.00 & 6.50 & 7.25 & 8.25 \
\hline y & 28 & 33 & 20 & 70 & 58 \
\hline
\end{array}$$ What percentage of the variation in y cannot be explained by the corresponding variation in x and the least-squares line?
A)25.5%
B)44.4%
C)55.6%
D)19.7%
E)74.5%

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: $$\begin{array} { c | c c c c c } 
\hline x & 5.75 & 6.00 & 6.50 & 7.25 & 8.25 \
\hline y & 28 & 33 & 20 & 70 & 58 \
\hline
\end{array}$$ What percentage of the variation in y cannot be explained by the corresponding variation in x and the least-squares line?
A)25.5%
B)44.4%
C)55.6%
D)19.7%
E)74.5%

Quizplus · Accepted Answer

The percentage of variation in y that cannot be explained by x is given by 1 - r², where r is the correlation coefficient. If 55.6% of the variation is explained by the least-squares line, then 44.4% is not explained.

It Is Thought That Prehistoric Indians Did Not Take Their x5.756.006.507.258.25y2833207058\begin{array} { c | c c c c c } \hline x & 5.75 & 6.00 & 6.50 & 7.25 & 8.25 \\\hline y & 28 & 33 & 20 & 70 & 58 \\\hline\end{array}xy​5.7528​6.0033​6.5020​7.2570​8.2558​​

It Is Thought That Prehistoric Indians Did Not Take Their $\begin{array} { c | c c c c c } \hline x & 5.75 & 6.00 & 6.50 & 7.25 & 8.25 \\\hline y & 28 & 33 & 20 & 70 & 58 \\\hline\end{array}$