To find the linear regression line [tex]\(\hat{y} = ax + b\)[/tex] for the given data points, we need to determine the slope [tex]\(a\)[/tex] and the intercept [tex]\(b\)[/tex]. We will follow these steps:
1. Calculate the means of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
\bar{x} \text{ (mean of } x \text{ values)} = 9.429
\][/tex]
[tex]\[
\bar{y} \text{ (mean of } y \text{ values)} = 74.671
\][/tex]
2. Calculate the terms needed for the numerator of the slope [tex]\(a\)[/tex]:
[tex]\[
\text{Numerator} = \sum (x_i - \bar{x})(y_i - \bar{y}) = -470.314
\][/tex]
3. Calculate the terms needed for the denominator of the slope [tex]\(a\)[/tex]:
[tex]\[
\text{Denominator} = \sum (x_i - \bar{x})^2 = 233.714
\][/tex]
4. Calculate the slope [tex]\(a\)[/tex]:
[tex]\[
a = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} = \frac{-470.314}{233.714} = -2.012
\][/tex]
5. Calculate the intercept [tex]\(b\)[/tex]:
[tex]\[
b = \bar{y} - a \cdot \bar{x} = 74.671 - (-2.012 \cdot 9.429) = 93.645
\][/tex]
Thus, the linear regression line that describes the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is:
[tex]\[
\boxed{\hat{y} = -2.012x + 93.645}
\][/tex]
The slope [tex]\(a\)[/tex] is [tex]\(-2.012\)[/tex] and the intercept [tex]\(b\)[/tex] is [tex]\(93.645\)[/tex], both rounded to the nearest thousandth.
