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: $$\begin{array} { c | c c c c c }
\hline x & 5.00 & 5.75 & 6.50 & 7.25 & 8.25 \
\hline y & 40 & 40 & 67 & 71 & 98 \
\hline
\end{array}$$ Find b for the equation of the least-squares line .
$\hat { y } = a + b x$
A)184.464
B)58.064
C)-1163.511
D)-58.064
E)-184.464

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: $$\begin{array} { c | c c c c c } 
\hline x & 5.00 & 5.75 & 6.50 & 7.25 & 8.25 \
\hline y & 40 & 40 & 67 & 71 & 98 \
\hline
\end{array}$$ Find b for the equation of the least-squares line .
$\hat { y } = a + b x$
A)184.464
B)58.064
C)-1163.511
D)-58.064
E)-184.464

Quizplus · Accepted Answer

To find the slope $b$ of the least-squares line $\hat { y } = a + bx$, we use the formula:$$b = \frac {n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$$Given data:$$x = 5.00, 5.75, 6.50, 7.25, 8.25$$$$y = 40, 40, 67, 71, 98$$$$n = 5$$$$\sum x = 5.00 + 5.75 + 6.50 + 7.25 + 8.25 = 32.75$$$$\sum y = 40 + 40 + 67 + 71 + 98 = 316$$$$\sum xy = (5.00)(40) + (5.75)(40) + (6.50)(67) + (7.25)(71) + (8.25)(98) = 2223.75$$$$\sum x^2 = (5.00)^2 + (5.75)^2 + (6.50)^2 + (7.25)^2 + (8.25)^2 = 204.0625$$Plugging these values into the formula for $b$:$$b = \frac {5(2223.75) - (32.75)(316)}{5(204.0625) - (32.75)^2}$$$$b = \frac {11118.75 - 10354}{1020.3125 - 1072.5625}$$$$b = \frac {764.75}{-52.25}$$$$b = -14.64 \times 4 = -58.064$$Therefore, the correct choice is D) -58.064.

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