The Complete Second-Order Model Was Fit To $n = 25$ Data Points

question 116

Essay

The complete second-order model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } x _ { 2 } + \beta _ { 4 } x _ { 1 } ^ { 2 } + \beta _ { 5 } x _ { 2 } ^ { 2 }$ was fit to $n = 25$ data points. The printout is shown below.
ANOVA
$\begin{array} { l l l l l l } & d f & \text { SS } & \text { MS } & F & \text { Significance } F \\\hline \text { Regression } & 5 & 22812.46538 & 4562.493077 & 56487.98 & 6.12671 \mathrm { E } - 39 \\\text { Residual } & 19 & 1.534616187 & 0.080769273 & & \\\text { Total } & 24 & 22814 & & & \\\hline\end{array}$

$\begin{array}{lllllll} & \text { Coefficients } & \text { Standard Error } & t \text { Stat } & \text { P-value } & \text { Lower 95\% } & \text { Upper 95\% } \\\hline \text { Intercept } & -0.202274307 & 0.377603882 & -0.535678569 & 0.598396064 & -0.99260856 & 0.588059946 \\\mathrm{X} 1 & 0.57956491 & 0.184697537 & 3.137913578 & 0.005416889 & 0.192988402 & 0.966141418 \\\mathrm{X} 2 & 0.502983937 & 0.130940123 & 3.841327815 & 0.001100855 & 0.228923024 & 0.777044849 \\\mathrm{Xl}^{*} \mathrm{X} 2 & 1.976110807 & 0.022011043 & 89.77815357 & 1.92982 \mathrm{E}-26 & 1.93004115 & 2.022180464 \\\mathrm{X} 1^{\wedge} 2 & -0.026825292 & 0.025350994 & -1.058155454 & 0.303252905 & -0.079885548 & 0.026234964 \\\mathrm{X}^{\wedge} 2 & 0.012944358 & 0.015088978 & 0.857868446 & 0.401657492 & -0.018637245 & 0.044525961\end{array}$

a. Write the complete second-order model for the data.
b. Is there sufficient evidence to indicate that at least one of the parameters $\beta _ { 1 } , \beta _ { 2 } , \beta _ { 3 } , \beta _ { 4 }$ , and $\beta 5$ is nonzero? Test using $\alpha = .05$ .
c. Test $H _ { 0 } : \beta _ { 3 } = 0$ against $H _ { \mathrm { a } } : \beta _ { 3 } \neq 0$ . Use $\alpha = .01$ .
d. Test $H _ { 0 } : \beta _ { 4 } = 0$ against $H _ { \mathrm { a } } : \beta _ { 4 } \neq 0$ . Use $\alpha = .01$ .

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