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:

P-Value

A statistical measure that helps scientists determine the significance of their research results.

Retention Rate

A metric usually used to measure the percentage of individuals or employees who remain engaged with an institution, company, or program over a given period.

Random Sample

A sample drawn in such a way that every member of the population has an equal chance of being selected.

Null Hypothesis

A statement used in statistical testing, which proposes that no significant difference exists between certain characteristics of a population.