This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
1. The least-squares regression equation is y = b_0 + b_1 x_1 + b_2 x_2 + b_3 x_3 where b_0 = 3.47, b_1 = -0.0042, b_2 = 0.012, b_3 = 0.10. Substitute the values: y = 3.47 + (-0.0042) x_1 + 0.012 x_2 + 0.10 x_3 y = 3.47 - 0.0042 x_1 + 0.012 x_2 + 0.10 x_3 y = 3.47 - 0.0042 x_1 + 0.012 x_2 + 0.10 x_3 2. The p-value for x_1 is 0.002. Since 0.002 < 0.05, the relationship between the response and x_1 is significant. The t-value is -3.07, and |t| = 3.07 > 2 also indicates significance. Yes, since P = 0.002 < 0.05 3. The slope for x_1 is b_1 = -0.0042 < 0. Thus, when x_1 increases by 1 unit (with x_2 and x_3 held constant), the mean response decreases by 0.0042 units. decreases by 0.0042 units per 1-unit increase in x_1 4. Substitute x_1 = 50, x_2 = 3, x_3 = 0.5 into the equation: y = 3.47 - 0.0042(50) + 0.012(3) + 0.10(0.5) Compute x_1 term: -0.0042 × 50 = -0.21 Compute x_2 term: 0.012 × 3 = 0.036 Compute x_3 term: 0.10 × 0.5 = 0.05 Now add step-by-step: 3.47 - 0.21 = 3.26 3.26 + 0.036 = 3.296 3.296 + 0.05 = 3.346 Round to three decimal places (consistent with coefficient precision): y = 3.346 3.35