To determine the equation of the least-squares regression line based on the given statistical analysis from the regression output, let's follow these steps:
1. Identify the Dependent and Independent Variables:
- The dependent variable (y) is typically represented as the variable being predicted or explained. In this case, it is [tex]\(\log(\text{Area})\)[/tex].
- The independent variable (x) is the predictor or explanatory variable. In this case, it is [tex]\(\log(\text{Time})\)[/tex].
2. Extract Key Coefficients from the Regression Output:
- The constant term (intercept) is given as 0.490.
- The coefficient for [tex]\(\log(\text{Time})\)[/tex] (slope) is given as 2.004.
3. Formulate the Regression Equation:
- The general form of a linear regression equation is:
[tex]\[
\hat{y} = b_0 + b_1 x
\][/tex]
where [tex]\( \hat{y} \)[/tex] is the predicted value, [tex]\( b_0 \)[/tex] is the intercept, and [tex]\( b_1 \)[/tex] is the slope.
4. Substitute the Given Values:
- Here, [tex]\( y = \log(\text{Area}) \)[/tex]
- [tex]\( x = \log(\text{Time}) \)[/tex]
- [tex]\( b_0 = 0.490 \)[/tex]
- [tex]\( b_1 = 2.004 \)[/tex]
Therefore, the equation becomes:
[tex]\[
\log(\text{Area}) = 0.490 + 2.004 \log(\text{Time})
\][/tex]
This detailed step-by-step analysis leads us to the equation of the least-squares regression line. Hence, the correct equation is:
[tex]\[
\log(\text{Area}) = 0.490 + 2.004 \log(\text{Time})
\][/tex]
