Let's determine the regression line equation [tex]\( y = a x + b \)[/tex] and the corresponding correlation coefficient [tex]\( r \)[/tex] using the given data. Here's the step-by-step solution:
1. Given Data:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
x & 138 & 104 & 131 & 135 & 116 & 92 & 103 \\
\hline
y & 15 & 4 & 8 & 10 & 7 & 3 & 4 \\
\hline
\end{array}
\][/tex]
2. Perform Linear Regression Analysis:
- To find the slope [tex]\( a \)[/tex] and the y-intercept [tex]\( b \)[/tex] of the regression line, we'll use the linear regression formulas.
- The correlation coefficient [tex]\( r \)[/tex] quantifies the strength and direction of the linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
3. Solution:
- After calculating the necessary statistics from the data (formulas include sums of products, means, etc.), we derive:
- The slope [tex]\( a \)[/tex] of the regression line,
- The intercept [tex]\( b \)[/tex], and
- The correlation coefficient [tex]\( r \)[/tex].
4. Values Calculation:
- Slope [tex]\( a \)[/tex]:
[tex]\[
a = 0.214
\][/tex]
- Intercept [tex]\( b \)[/tex]:
[tex]\[
b = -17.709
\][/tex]
- Correlation Coefficient [tex]\( r \)[/tex]:
[tex]\[
r = 0.911
\][/tex]
5. Regression Line Equation:
Plugging the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the regression line equation:
[tex]\[
y = 0.214 x - 17.709
\][/tex]
Therefore, the regression line equation is:
[tex]\[
y = 0.214 x - 17.709
\][/tex]
And the corresponding correlation coefficient is:
[tex]\[
r = 0.911
\][/tex]
These values are rounded to the nearest thousandth as required.
