Use the given data to find the equation of the regression line. Examine the scatterplot and identify a characteristic of the data that is ignored by the regression line.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline \begin{tabular}{cc}
$x$ & 12 \\
$y$ & 11.86 \\
\end{tabular} & \begin{tabular}{c}
10 \\
11.38
\end{tabular} & \begin{tabular}{c}
11 \\
11.80
\end{tabular} & \begin{tabular}{c}
9 \\
10.60
\end{tabular} & \begin{tabular}{c}
6 \\
6.14
\end{tabular} & \begin{tabular}{c}
13 \\
11.58
\end{tabular} & \begin{tabular}{c}
8 \\
9.46
\end{tabular} & \begin{tabular}{c}
5 \\
3.94
\end{tabular} & \begin{tabular}{c}
14 \\
10.94
\end{tabular} & \begin{tabular}{c}
7 \\
7.98
\end{tabular} & \begin{tabular}{c}
15 \\
9.94
\end{tabular} \\
\hline
\end{tabular}
$\hat{y}=\square+\square \times$ (Round to two decimal places as needed.)
Answer
The characteristic of the data that is ignored by the regression line is the individual variation of the data points around the line. The regression line is a best fit line and does not pass through all the data points. Therefore, it does not capture the individual variation of the data points.
Step 1 ：The given data points are (12, 11.86), (10, 11.38), (11, 11.80), (9, 10.60), (6, 6.14), (13, 11.58), (8, 9.46), (5, 3.94), (14, 10.94), (7, 7.98), (15, 9.94).
Step 2 ：The regression line is a line that best fits the data points in a scatter plot. It is used to predict the value of one variable based on the value of another variable.
Step 3 ：The equation of the regression line is given by \(y = mx + c\), where \(m\) is the slope of the line and \(c\) is the y-intercept.
Step 4 ：The slope \(m\) is given by the formula \((nΣxy - ΣxΣy) / (nΣx^2 - (Σx)^2)\) and the y-intercept \(c\) is given by the formula \((Σy - mΣx) / n\), where \(n\) is the number of data points, \(Σxy\) is the sum of the product of \(x\) and \(y\), \(Σx\) is the sum of \(x\), \(Σy\) is the sum of \(y\), and \(Σx^2\) is the sum of the square of \(x\).
Step 5 ：Using the given data, we find that \(n = 11\), \(Σx = 110\), \(Σy = 105.62\), \(Σxy = 1122.20\), and \(Σx^2 = 1210\).
Step 6 ：Substituting these values into the formulas, we find that \(m = 0.60\) and \(c = 3.60\).
Step 7 ：Substituting these values into the equation of the regression line, we get \(\hat{y} = 0.60x + 3.60\).
Step 8 ：Final Answer: The equation of the regression line is \(\hat{y} = \boxed{0.60}x + \boxed{3.60}\) (rounded to two decimal places).
Step 9 ：The characteristic of the data that is ignored by the regression line is the individual variation of the data points around the line. The regression line is a best fit line and does not pass through all the data points. Therefore, it does not capture the individual variation of the data points.
