To find the equation of the least-squares regression line for predicting acrylamide concentration ([tex]\(\hat{y}\)[/tex]) using frying time ([tex]\(x\)[/tex]), we need to determine the slope and intercept of the line. The equation of the line is:
[tex]\[
\hat{y} = mx + b
\][/tex]
Where:
- [tex]\(m\)[/tex] is the slope of the line
- [tex]\(b\)[/tex] is the y-intercept of the line
- [tex]\(x\)[/tex] is the independent variable (frying time)
- [tex]\(\hat{y}\)[/tex] is the dependent variable (acrylamide concentration)
Given the data:
[tex]\[
\begin{array}{|c|c|}
\hline \text{Frying Time (seconds)} & \text{Acrylamide Concentration (micrograms/kg)} \\
\hline
150 & 150 \\
240 & 115 \\
240 & 190 \\
270 & 185 \\
300 & 145 \\
300 & 270 \\
\hline
\end{array}
\][/tex]
From the analysis, the slope ([tex]\(m\)[/tex]) and y-intercept ([tex]\(b\)[/tex]) are calculated as:
- Slope ([tex]\(m\)[/tex]): 0.4103
- Intercept ([tex]\(b\)[/tex]): 73.2692
Thus, the equation of the least-squares regression line is:
[tex]\[
\hat{y} = 0.4103x + 73.2692
\][/tex]
This equation can be used to predict the acrylamide concentration based on the frying time. Just plug in the values for [tex]\(x\)[/tex] (frying time) into the equation to find [tex]\(\hat{y}\)[/tex] (predicted acrylamide concentration).
