To find the equation of the regression line for the given data, we should follow these steps:
1. Calculate the Mean of Productivity and Dexterity:
[tex]\[
\text{Mean Productivity} (\bar{x}) = \frac{23 + 25 + 28 + 21 + 21 + 25 + 26 + 30 + 34 + 36}{10} = 26.9
\][/tex]
[tex]\[
\text{Mean Dexterity} (\bar{y}) = \frac{49 + 53 + 59 + 42 + 47 + 53 + 55 + 63 + 67 + 75}{10} = 56.3
\][/tex]
2. Calculate the Slope (b1) for the Regression Line:
The slope [tex]\( b_1 \)[/tex] is given by:
[tex]\[
b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\][/tex]
Given the values, this calculation has been determined as:
[tex]\[
b_1 = 1.905
\][/tex]
3. Calculate the Intercept (b0) for the Regression Line:
The intercept [tex]\( b_0 \)[/tex] is given by:
[tex]\[
b_0 = \bar{y} - b_1 \cdot \bar{x}
\][/tex]
Using the values:
[tex]\[
b_0 = 56.3 - (1.905 \cdot 26.9) = 5.055
\][/tex]
4. Formulate the Regression Line Equation:
The equation of the regression line is:
[tex]\[
\hat{y} = b_0 + b_1 x
\][/tex]
Plugging in the calculated values:
[tex]\[
\hat{y} = 5.055 + 1.905 x
\][/tex]
Rounding to three significant digits where necessary, we have:
[tex]\(
\boxed{\hat{y} = 5.05 + 1.91x}
\)[/tex]
Thus, the correct regression equation is:
[tex]\(
\boxed{\hat{y} = 5.05 + 1.91x}
\)[/tex]
