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15.19 Acrylamide is a chemical that is sometimes found in cooked starchy foods and which is thought to increase the risk of certain kinds of cancer. The paper "A Statistical Regression Model for the Estimation of Acrylamide Concentrations in French Fries for Excess Lifetime Cancer Risk Assessment" (Food and Chemical Toxicology [2012]: \(3867-3876\) ) describes a study to investigate the effect of frying time (in seconds) and acrylamide concentration (in micrograms per kilogram) in french fries. The data in the accompanying table are approximate values read from a graph that appeared in the paper. \begin{tabular}{|cc|} \hline Frying Time & Acrylamide Concentration \\ \hline 150 & 155 \\ 240 & 120 \\ 240 & 190 \\ 270 & 185 \\ 300 & 140 \\ 300 & 270 \\ \hline \end{tabular} a. For these data, the estimated regression line for predicting \(y=\) acrylamide concentration based on \(x=\) frying time is \(y=87+0.359 x\). What is an estimate of the average change in acrylamide concentration associated with a 1-second increase in frying time? b. What would you predict for acrylamide concentration for a frying time of 250 seconds? c. Use the given Minitab output to decide if there is convincing evidence of a useful linear relationship between acrylamide concentration and frying time. You may assume that the necessary conditions have been met. R-sq \(\begin{array}{cc}\text { R-sq(adj) } & \text { R-sq(pred) } \\ 0.00 \% & 0.00 \%\end{array}\) \(\mathrm{S}\) 3 \(\mathrm{q}\) \(8 \%\) Coefficients \(\mathrm{K}-\mathrm{Sq}\) \(14.38 \%\) \(\begin{array}{lccccc}\text { Term } & \text { Coef } & \text { SE Coef } & \text { T-Value } & \text { P-Value } & \text { VIF } \\ \text { Constant } & 87 & 112 & 0.78 & 0.480 & \\ x & 0.359 & 0.438 & 0.82 & 0.459 & 1.00\end{array}\) Regression Equation \(y=87+0.359 x\)

Step by step solution

01

Calculate the average change in acrylamide concentration per 1-second increase in frying time

From the regression equation, y = 87 + 0.359 * x, the coefficient of x represents the average change in acrylamide concentration with respect to frying time. Therefore, the estimated average change in acrylamide concentration associated with a 1-second increase in frying time is 0.359 micrograms per kilogram.

02

Predict acrylamide concentration for a frying time of 250 seconds

Using the given regression equation y = 87 + 0.359x, we substitute x with the given frying time of 250 seconds. y = 87 + (0.359 * 250) = 87 + 89.75 = 176.75 Thus, the predicted acrylamide concentration for a frying time of 250 seconds is 176.75 micrograms per kilogram.

03

Determine if there is convincing evidence of a linear relationship between acrylamide concentration and frying time

We will use the Minitab output provided to evaluate the evidence of a linear relationship. Here are the key statistics to look at: 1. R-squared: The R-squared value is 0.00%, which indicates that none of the variability in acrylamide concentration is explained by the linear relationship with frying time. 2. P-value: The P-value for x (frying time) is 0.459, which is greater than the common significance level of 0.05. This means that we fail to reject the null hypothesis and there is not enough evidence to suggest a useful linear relationship between frying time and acrylamide concentration. Based on the provided Minitab output, there is no convincing evidence of a useful linear relationship between acrylamide concentration and frying time. The low R-squared value and high P-value indicate that the regression equation does not explain the variability in acrylamide concentration well.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Acrylamide Concentration Analysis

Understanding acrylamide concentration in food items, notably in starchy foods like french fries, is critical due to potential health risks. Acrylamide is a chemical that may form during high-temperature cooking processes, such as frying, and extensive research suggest it could increase cancer risk.

Analysis of acrylamide concentration typically involves collecting data from various food samples under different conditions, such as frying time or temperature, and measuring the acrylamide level present in micrograms per kilogram. In the educational context of the given problem, students are looking at approximate data values from a research paper that attempts to determine the relationship between frying time and acrylamide concentration in french fries.

To gain insights from this data, statistical methods are applied, and the focus is often on establishing whether there is a consistent pattern that can be modeled. For instance, does a longer frying time consistently result in a higher concentration of acrylamide? Answering such questions requires adopting a rigorous analytical approach, which often takes the form of regression analysis—a statistical method to understand the relationship between variables.

Linear Regression Analysis

Linear regression analysis is a powerful statistical tool used to model the relationship between a dependent variable and one or more independent variables. The main goal is to find the linear equation that best predicts the dependent variable from the independent variables.

In our example, the dependent variable is the acrylamide concentration in french fries, while the independent variable is the frying time. The equation of the form y = a + bx is proposed, where y represents the acrylamide concentration, x is the frying time, a is the intercept, and b is the slope of the line.

The slope b is particularly important as it indicates the average change in the dependent variable for a unit change in the independent variable. A positive slope suggests that as frying time increases, so does the acrylamide concentration. Conversely, a negative slope would indicate that longer frying times are associated with lower acrylamide concentrations. In this case, the slope of 0.359 implies that for every additional second of frying, the acrylamide concentration increases by 0.359 micrograms per kilogram on average.

Regression Equation Interpretation

Interpreting the regression equation is essential for drawing meaningful conclusions from data. The equation represents a model that predicts the estimated value of the dependent variable based on the independent variable(s).

For the given example, the regression equation y=87+0.359x enables one to predict the acrylamide concentration in french fries based on frying time. The number 87 represents the estimated acrylamide concentration when frying time is zero. The coefficient 0.359 reflects the estimated change in acrylamide concentration with each additional second of frying.

However, just having a regression equation isn't enough. One must evaluate its statistical significance and the strength of the linear relationship it depicts. The indicators for this are the R-squared value and P-value. The former measures how well the independent variable(s) explain the variability of the dependent variable, whilst the latter tests the hypothesis of no relationship. In this case, the R-squared value of 0.00% and a high P-value suggest that the frying time isn't a significant predictor of acrylamide concentration, leading us to conclude that the equation doesn't provide a reliable prediction model in this context.