To find the line of least squares regression for the given data, we can use the method of linear regression to derive the equation of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Using the data provided:
\[ \begin{array}{c|c} x & y \ \hline 0 & 8 \ 4 & 9 \ 8 & 10 \ 12 & 11 \ 16 & 12 \ \end{array} \]
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Calculate the means of x and y:
\( \bar{x} = \frac{0 + 4 + 8 + 12 + 16}{5} = \frac{40}{5} = 8 \)
\( \bar{y} = \frac{8 + 9 + 10 + 11 + 12}{5} = \frac{50}{5} = 10 \)
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Calculate \( m \) (the slope):
\[ m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]
- Calculate the necessary values:
\[ \begin{array}{c|c|c|c|c} x & y & x - \bar{x} & y - \bar{y} & (x - \bar{x})(y - \bar{y}) \ \hline 0 & 8 & 0 - 8 = -8 & 8 - 10 = -2 & 16 \ 4 & 9 & 4 - 8 = -4 & 9 - 10 = -1 & 4 \ 8 & 10 & 8 - 8 = 0 & 10 - 10 = 0 & 0 \ 12 & 11 & 12 - 8 = 4 & 11 - 10 = 1 & 4 \ 16 & 12 & 16 - 8 = 8 & 12 - 10 = 2 & 16 \ \end{array} \]
- Sum of products:
\[ \sum (x_i - \bar{x})(y_i - \bar{y}) = 16 + 4 + 0 + 4 + 16 = 40 \]
- Sum of squares:
\[ \sum (x_i - \bar{x})^2 = (-8)^2 + (-4)^2 + 0^2 + 4^2 + 8^2 = 64 + 16 + 0 + 16 + 64 = 160 \]
- Calculate slope \( m \):
\[ m = \frac{40}{160} = 0.25 \]
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Calculate \( b \) (the y-intercept):
\[ b = \bar{y} - m\bar{x} = 10 - 0.25 \cdot 8 = 10 - 2 = 8 \]
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Final equation:
Thus, the equation of the regression line is:
\[ y = 0.25x + 8 \]
Therefore, the correct response is:
y = 0.25x + 8.
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