Chapter 12, Problem 12.136LM

Suppose you fit the model

     $y\text{ =}{\beta }_{\text{0}}\text{+ }{\beta }_{\text{1}}{x}_{\text{1}}\text{ + }{\beta }_2{x}_1^2\text{ + }{\beta }_3{x}_{\text{2}}\text{ + }{\beta }_4{x}_{\text{1}}{x}_{\text{2}}\text{ + }\epsilon$

to n = 25 data points with the following results:

     ${\hat{\beta }}_{\text{0}}=1.26{\hat{\beta }}_1=-2.43\text{}\text{}\text{}{\hat{\beta }}_{\text{2}}=0.5{\hat{\beta }}_{\text{3}}=.62{\hat{\beta }}_{\text{4}}=1.81$

     ${\text{s}}_{{\hat{\beta }}_1}\text{ = 1}{\text{.21 s}}_{{\hat{\beta }}_2}\text{ = }{\text{.16 s}}_{{\hat{\beta }}_3}\text{ = }{\text{.26 s}}_{{\hat{\beta }}_4}\text{ = 1}\text{.49}$

     $\text{SSE = }\text{.41 }{R}^{\text{2}}\text{ = }\text{.83}$

1. a. Is there sufficient evidence to conclude that at least one of the parameters β₁, β₂, β₃, or β₄ is nonzero? Test using α=.05.
2. b. Test H₀: β₁ = 0 against H_a: β₁ < 0. Use α =.05.
3. c. Test H₀: β₂ = 0 against H_a: β₂ > 0. Use α =.05.
4. d. Test H₀: β₃ = 0 against H_a: β₃ ≠ 0. Use α =.05.