To find the equation of the regression line using the given data on performance and attitude ratings, we follow these steps:
1. Calculate the Means:
- Mean of performance scores: [tex]\( \bar{x} = 67.0 \)[/tex]
- Mean of attitude scores: [tex]\( \bar{y} = 80.1 \)[/tex]
2. Calculate the Numerator and Denominator for the Slope [tex]\( b_1 \)[/tex] of the Regression Line:
- Numerator: [tex]\( \sum{(x_i - \bar{x})(y_i - \bar{y})} = 380.0 \)[/tex]
- Denominator: [tex]\( \sum{(x_i - \bar{x})^2} = 372.0 \)[/tex]
3. Calculate the Slope [tex]\( b_1 \)[/tex]:
[tex]\[
b_1 = \frac{\text{Numerator}}{\text{Denominator}} = \frac{380.0}{372.0} = 1.021505376344086 \approx 1.02
\][/tex]
4. Calculate the Intercept [tex]\( b_0 \)[/tex]:
[tex]\[
b_0 = \bar{y} - b_1 \bar{x} = 80.1 - (1.021505376344086 \times 67.0) = 11.659139784946234 \approx 11.7
\][/tex]
5. Construct the Regression Line Equation:
- Using [tex]\( b_0 \approx 11.7 \)[/tex] and [tex]\( b_1 \approx 1.02 \)[/tex], the equation of the regression line is:
[tex]\[
\hat{y} = 11.7 + 1.02x
\][/tex]
Given the final options, the correct regression equation is:
[tex]\[
\boxed{\hat{y} = 11.7 + 1.02x}
\][/tex]
