Answer :
## Part (a) Interpretation of [tex]\( b_1 \)[/tex] and [tex]\( b_2 \)[/tex]
In the given regression equation [tex]\( y = 29.1031 + 0.5549 x_1 + 0.4097 x_2 \)[/tex]:
- [tex]\( b_1 = 0.5549 \)[/tex]:
- This coefficient indicates that for every 1-unit increase in [tex]\( x_1 \)[/tex], the value of [tex]\( y \)[/tex] is expected to increase by 0.5549 units, provided that [tex]\( x_2 \)[/tex] remains constant.
- [tex]\( b_2 = 0.4097 \)[/tex]:
- This coefficient indicates that for every 1-unit increase in [tex]\( x_2 \)[/tex], the value of [tex]\( y \)[/tex] is expected to increase by 0.4097 units, provided that [tex]\( x_1 \)[/tex] remains constant.
## Part (b) Estimate [tex]\( y \)[/tex] when [tex]\( x_1 = 180 \)[/tex] and [tex]\( x_2 = 310 \)[/tex]
To estimate [tex]\( y \)[/tex] for [tex]\( x_1 = 180 \)[/tex] and [tex]\( x_2 = 310 \)[/tex], we substitute these values into the regression equation:
[tex]\[ y = 29.1031 + 0.5549 \cdot 180 + 0.4097 \cdot 310 \][/tex]
Let's break down the computation step-by-step:
1. Calculate [tex]\( 0.5549 \cdot 180 \)[/tex]:
[tex]\[ 0.5549 \cdot 180 = 99.882 \][/tex]
2. Calculate [tex]\( 0.4097 \cdot 310 \)[/tex]:
[tex]\[ 0.4097 \cdot 310 = 126.007 \][/tex]
3. Add these values to the constant term [tex]\( 29.1031 \)[/tex]:
[tex]\[ y = 29.1031 + 99.882 + 126.007 \][/tex]
[tex]\[ y = 255.9921 \][/tex]
Thus, the estimated value of [tex]\( y \)[/tex] when [tex]\( x_1 = 180 \)[/tex] and [tex]\( x_2 = 310 \)[/tex] is:
[tex]\[ y \approx 255.992 \][/tex]
In the given regression equation [tex]\( y = 29.1031 + 0.5549 x_1 + 0.4097 x_2 \)[/tex]:
- [tex]\( b_1 = 0.5549 \)[/tex]:
- This coefficient indicates that for every 1-unit increase in [tex]\( x_1 \)[/tex], the value of [tex]\( y \)[/tex] is expected to increase by 0.5549 units, provided that [tex]\( x_2 \)[/tex] remains constant.
- [tex]\( b_2 = 0.4097 \)[/tex]:
- This coefficient indicates that for every 1-unit increase in [tex]\( x_2 \)[/tex], the value of [tex]\( y \)[/tex] is expected to increase by 0.4097 units, provided that [tex]\( x_1 \)[/tex] remains constant.
## Part (b) Estimate [tex]\( y \)[/tex] when [tex]\( x_1 = 180 \)[/tex] and [tex]\( x_2 = 310 \)[/tex]
To estimate [tex]\( y \)[/tex] for [tex]\( x_1 = 180 \)[/tex] and [tex]\( x_2 = 310 \)[/tex], we substitute these values into the regression equation:
[tex]\[ y = 29.1031 + 0.5549 \cdot 180 + 0.4097 \cdot 310 \][/tex]
Let's break down the computation step-by-step:
1. Calculate [tex]\( 0.5549 \cdot 180 \)[/tex]:
[tex]\[ 0.5549 \cdot 180 = 99.882 \][/tex]
2. Calculate [tex]\( 0.4097 \cdot 310 \)[/tex]:
[tex]\[ 0.4097 \cdot 310 = 126.007 \][/tex]
3. Add these values to the constant term [tex]\( 29.1031 \)[/tex]:
[tex]\[ y = 29.1031 + 99.882 + 126.007 \][/tex]
[tex]\[ y = 255.9921 \][/tex]
Thus, the estimated value of [tex]\( y \)[/tex] when [tex]\( x_1 = 180 \)[/tex] and [tex]\( x_2 = 310 \)[/tex] is:
[tex]\[ y \approx 255.992 \][/tex]
