An Economist Is in the Process of Developing a Model $\left( x _ { 1 } \right)$

Essay

An economist is in the process of developing a model to predict the price of gold. She believes that the two most important variables are the price of a barrel of oil $\left( x _ { 1 } \right)$ and the interest rate $\left( x _ { 2 } \right)$ She proposes the first-order model with interaction: $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } x _ { 3 } + \varepsilon$ .
A random sample of 20 daily observations was taken. The computer output is shown below.
THE REGRESSION EQUATION IS $y =$ $115.6 + 22.3 x _ { 1 } + 14.7 x _ { 2 } - 1.36 x _ { 1 } x _ { 2 }$ . $$\begin{array} { | c | r r r | } \hline \text { Predictor } & \text { Coef } & \text { StDev } & T \\\hline \text { Constant } & 115.6 & 78.1 & 1.480 \\x _ { 1 } & 22.3 & 7.1 & 3.141 \\x _ { 2 } & 14.7 & 6.3 & 2.333 \\x _ { 1 } x _ { 2 } & - 1.36 & 0.52 & - 2.615 \\\hline\end{array}$$ S = 20.9 R-Sq = 55.4%. $$\begin{array}{l}\text { ANALYSIS OF VARIANCE }\\\begin{array} { | l | r r r r | } \hline \text { Source of Variation } & d f & \text { SS } & \text { MS } & F \\\hline \text { Regression } & 3 & 8661 & 2887.0 & 6.626 \\\text { Error } & 16 & 6971 & 435.7 & \\\hline \text { Total } & 19 & 15632 & & \\\hline\end{array}\end{array}$$ Do these results allow us at the 5% significance level to conclude that the model is useful in predicting the price of gold?