\begin{tabular}{|l|c|c|c|c|c|c|} \hline Height, $x$ & 775 & 619 & 519 & 508 & 491 & 474 \\ \hline Stories, $y$ & 53 & 47 & 46 & 41 & 38 & 36 \\ \hline \end{tabular} (a) $x=502$ feet (b) $x=650$ feet (c) $x=810$ feet (d) $x=731$ feet Find the regression equation. \[ \hat{y}=\square x+ \] (Round the slope to three decimal places as needed. Round the $y$-intercept to two decimal places as needed.)
Solution
Step 1 ：The question is asking for the regression equation of the given data. The regression equation is a statistical method used to describe the relationship between two variables. In this case, the two variables are the height of a building (x) and the number of stories in the building (y). The regression equation is usually in the form of \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.
Step 2 ：To find the regression equation, we need to calculate the slope (\(m\)) and the y-intercept (\(c\)). The slope can be calculated using the formula: \[m = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2}\] And the y-intercept can be calculated using the formula: \[c = \frac{\Sigma y - m(\Sigma x)}{n}\]
Step 3 ：Where: \(n\) is the number of data points, \(\Sigma xy\) is the sum of the product of x and y, \(\Sigma x\) is the sum of x, \(\Sigma y\) is the sum of y, \(\Sigma x^2\) is the sum of x squared.
Step 4 ：Given the data points: \[x = [775, 619, 519, 508, 491, 474]\] \[y = [53, 47, 46, 41, 38, 36]\] We have: \[n = 6\] \[\Sigma x = 3386\] \[\Sigma y = 261\] \[\Sigma xy = 150592\] \[\Sigma x^2 = 1976968\]
Step 5 ：Substitute these values into the formulas to calculate the slope and y-intercept: \[m = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2} = \frac{6(150592) - (3386)(261)}{6(1976968) - (3386)^2} = 0.0499128050562987\] \[c = \frac{\Sigma y - m(\Sigma x)}{n} = \frac{261 - 0.0499128050562987(3386)}{6} = 15.332540346562103\]
Step 6 ：Round the slope to three decimal places and the y-intercept to two decimal places, we get: \[m = 0.050\] \[c = 15.33\]
Step 7 ：Finally, substitute \(m\) and \(c\) into the regression equation \(\hat{y} = mx + c\), we get the final regression equation: \[\boxed{\hat{y} = 0.050x + 15.33}\]
