The Complete Second-Order Model Data Points $\text { ANOVA }$

question 77

Essay

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


$\begin{array}{lllllll}\hline & \text { Coefficients } & \text { Standard Error } & t \text { Stat } & \text { P-value } & \text { Lower 95\% } & \text { Upper 95\% } \\\hline \text { Intercept } & -0.202274307 & 0.377603882 & -0.535678569 & 0.598396064 & -0.99260856 & 0.588059946 \\\mathrm{X} 1 & 0.57956491 & 0.184697537 & 3.137913578 & 0.005416889 & 0.192988402 & 0.966141418 \\\mathrm{X} 2 & 0.502983937 & 0.130940123 & 3.841327815 & 0.001100855 & 0.228923024 & 0.777044849 \\\mathrm{X}1^{*} \mathrm{X} 2 & 1.976110807 & 0.022011043 & 89.77815357 & 1.92982 \mathrm{E}-26 & 1.93004115 & 2.022180464 \\\mathrm{X}1^{\wedge} 2 & -0.026825292 & 0.025350994 & -1.058155454 & 0.303252905 & -0.079885548 & 0.026234964 \\\mathrm{X}2^{\wedge} 2 & 0.012944358 & 0.015088978 & 0.857868446 & 0.401657492 & -0.018637245 & 0.044525961 \\\hline\end{array}$ 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 $\alpha = .01$ . 3 Test if Model is Useful for Predicting y

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Definitions:

Underwater Landslide

A movement of rock, earth, or debris down a slope in an underwater environment, often triggered by earthquakes or other disturbances.

Tsunami

Ocean waves of exceptionally long lengths triggered by significant disturbances such as seismic activity, volcanic explosions, or submarine landslides.

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The release of magma, gases, and ash from a volcano, which can form new landforms and impact climate.

Landslide

The movement of rock, earth, or debris down a slope due to gravity, often triggered by factors like rain, earthquakes, or human activity.