The Staff of a Test Kitchen Is Attempting to Determine $y$

question 92

Essay

The staff of a test kitchen is attempting to determine the baking time, $y$ , of a roast, i.e., the time it takes the internal temperature of the roast to reach $165 ^ { \circ } \mathrm { F }$ , using two variables, the temperature setting of the oven, $x _ { 1 }$ , and the weight of the roast, $x _ { 2 }$ , in pounds. The data for 24 roasts are shown below.
Baking Times of Roasts
$\begin{array}{lll|lll|lll|lll|lll|lll}\mathrm{X} 1\left({ }^{\circ} \mathrm{F}\right) & \mathrm{X} 2(\mathrm{lb}) & \mathrm{Y}(\mathrm{hr}) & \mathrm{X} 1\left({ }^{\circ} \mathrm{F}\right) & \mathrm{X} 2(\mathrm{lb}) & \mathrm{Y}(\mathrm{hr}) & \mathrm{X} 1\left({ }^{\circ} \mathrm{F}\right) & \mathrm{X} 2(\mathrm{lb})&\mathrm{Y}(\mathrm{hr}) & \mathrm{X} 1\left({ }^{\circ} \mathrm{F}\right) & \mathrm{X} 2(\mathrm{lb}) &\mathrm{Y}(\mathrm{hr}) \\\hline 300 & 2.2 & 2.6 & 325 & 2.1 & 2.3 & 350 & 2.3 & 2.2 & 375 & 2.2 & 1.9 \\300 & 2.7 & 2.8 & 325 & 2.4 & 2.4 & 350 & 2.5 & 2.3 & 375 & 2.6 & 2.2 \\300 & 2.9 & 3.1 & 325 & 2.9 & 2.6 & 350 & 2.8 & 2.5 & 375 & 2.9 & 2.4 \\300 & 3.1 & 3.2 & 325 & 3.0 & 2.7 & 350 & 3.2 & 2.7 & 375 & 3.1 & 2.6 \\300 & 3.2 & 3.2 & 325 & 3.2 & 2.9 & 350 & 3.4 & 2.8 & 375 & 3.3 & 2.7 \\300 & 3.5 & 3.3 & 325 & 3.6 & 3.1 & 350 & 3.5 & 2.8 & 375 & 3.4 & 2.7 \\\hline\end{array}$


a. Fit a complete second-order model to the data.
b. Do the data provide sufficient evidence to indicate that the second-order terms contribute information for the prediction of $y$ ? State the null and alternative hypotheses and the test statistic. Use $\alpha = .05$ .

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Definitions:

Monopoly Structure

A market structure characterized by a single seller that controls the whole industry, leading to a lack of competition and high prices for consumers.

Relevant Market

The market in which a particular product or service is sold, considering the competition and the demand of consumers.

Rule Of Reason

A legal doctrine that determines the legality of certain business practices based on their harm or benefit to competition.

Physical Activity

Any bodily movement produced by skeletal muscles that requires energy expenditure, contributing to health and fitness.