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"† describes a study to investigate the effect of x = frying time (in seconds) and y = acrylamide concentration (in micrograms per kg) in french fries. The data in the accompanying table are approximate values read from a graph that appeared in the paper.

Frying
Time Acrylamide
Concentration

150 155

240 120

240 195

270 185

300 145

300 270

(a)

Construct a scatterplot of these data.

A scatterplot has 6 points.

The horizontal axis is labeled "x" and ranges from 100 to 350.

The vertical axis is labeled "y" and ranges from 50 to 350.

The points are plotted from left to right in a downward, diagonal direction starting from the upper left of the diagram.

Along the horizontal axis, there are 2 points at 150, 1 point at 180, 2 points at 210, and 1 point at 300.

The points are scattered and are between the approximate horizontal axis values of 150 and 300 and between the approximate vertical values of 120 and 270.

A scatterplot has 6 points.

The horizontal axis is labeled "x" and ranges from 100 to 350.

The vertical axis is labeled "y" and ranges from 50 to 350.

The points are plotted from left to right in a horizontal direction starting from the lower left side of the diagram.

Along the horizontal axis, there is 1 point at 150, 2 points at 240, 1 point at 270, and 2 points at 300.

The points are scattered and are between the approximate horizontal axis values of 150 and 300 and between the approximate vertical values of 100 and 240.

A scatterplot has 6 points.

The horizontal axis is labeled "x" and ranges from 100 to 350.

The vertical axis is labeled "y" and ranges from 50 to 350.

The points are plotted from left to right in an upward, diagonal direction starting from the lower left of the diagram.

Along the horizontal axis, there is 1 point at 150, 2 points at 240, 1 point at 270, and 2 points at 300.

The points are somewhat scattered and are between the approximate horizontal axis values of 150 and 300 and between the approximate vertical values of 140 and 230.

A scatterplot has 6 points.

The horizontal axis is labeled "x" and ranges from 100 to 350.

The vertical axis is labeled "y" and ranges from 50 to 350.

The points are plotted from left to right in an upward, diagonal direction starting from the lower left of the diagram.

Along the horizontal axis, there is 1 point at 150, 2 points at 240, 1 point at 270, and 2 points at 300.

The points are scattered and are between the approximate horizontal axis values of 150 and 300 and between the approximate vertical values of 120 and 270.

(b)

Find the equation of the least squares regression line. (Round your answers to four decimal places.)

ŷ =  +



x

Based on this line, what would you predict acrylamide concentration (in micrograms per kg) to be for a frying time of 270 seconds? What is the residual (in micrograms per kg) associated with the observation (270, 185)? (Round your answers to two decimal places.)

predicted value micrograms per kgresidual micrograms per kg

(c)

Look again at the scatterplot from part (a). Which observation is potentially influential? Explain the reason for your choice. (Hint: See Example 4.9.)

The observation (270, 185) is potentially influential because that point has an x value that is far away from the rest of the data set.The observation (300, 145) is potentially influential because that point has an x value that is far away from the rest of the data set.    The observation (300, 145) is potentially influential because that point has a large residual.The observation (150, 155) is potentially influential because that point has a large residual.The observation (150, 155) is potentially influential because that point has an x value that is far away from the rest of the data set.

(d)

When the potentially influential observation is deleted from the data set, the equation of the least squares regression line fit to the remaining five observations is
ŷ = −42 + 0.83x.
Use this equation to predict acrylamide concentration (in micrograms per kg) for a frying time of 270 seconds.

micrograms per kg

Compare this prediction to the prediction made in part (b).

Question

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"† describes a study to investigate the effect of x = frying time (in seconds) and y = acrylamide concentration (in micrograms per kg) in french fries. The data in the accompanying table are approximate values read from a graph that appeared in the paper.

Frying Time	Acrylamide Concentration
150	155
240	120
240	195
270	185
300	145
300	270

(a)

Construct a scatterplot of these data.

A scatterplot has 6 points.

The horizontal axis is labeled "x" and ranges from 100 to 350.
The vertical axis is labeled "y" and ranges from 50 to 350.
The points are plotted from left to right in a downward, diagonal direction starting from the upper left of the diagram.
Along the horizontal axis, there are 2 points at 150, 1 point at 180, 2 points at 210, and 1 point at 300.
The points are scattered and are between the approximate horizontal axis values of 150 and 300 and between the approximate vertical values of 120 and 270.

A scatterplot has 6 points.

The horizontal axis is labeled "x" and ranges from 100 to 350.
The vertical axis is labeled "y" and ranges from 50 to 350.
The points are plotted from left to right in a horizontal direction starting from the lower left side of the diagram.
Along the horizontal axis, there is 1 point at 150, 2 points at 240, 1 point at 270, and 2 points at 300.
The points are scattered and are between the approximate horizontal axis values of 150 and 300 and between the approximate vertical values of 100 and 240.

A scatterplot has 6 points.

The horizontal axis is labeled "x" and ranges from 100 to 350.
The vertical axis is labeled "y" and ranges from 50 to 350.
The points are plotted from left to right in an upward, diagonal direction starting from the lower left of the diagram.
Along the horizontal axis, there is 1 point at 150, 2 points at 240, 1 point at 270, and 2 points at 300.
The points are somewhat scattered and are between the approximate horizontal axis values of 150 and 300 and between the approximate vertical values of 140 and 230.

A scatterplot has 6 points.

The horizontal axis is labeled "x" and ranges from 100 to 350.
The vertical axis is labeled "y" and ranges from 50 to 350.
The points are plotted from left to right in an upward, diagonal direction starting from the lower left of the diagram.
Along the horizontal axis, there is 1 point at 150, 2 points at 240, 1 point at 270, and 2 points at 300.
The points are scattered and are between the approximate horizontal axis values of 150 and 300 and between the approximate vertical values of 120 and 270.

(b)

Find the equation of the least squares regression line. (Round your answers to four decimal places.)

ŷ = +

x

Based on this line, what would you predict acrylamide concentration (in micrograms per kg) to be for a frying time of 270 seconds? What is the residual (in micrograms per kg) associated with the observation (270, 185)? (Round your answers to two decimal places.)

predicted value micrograms per kgresidual micrograms per kg

(c)

Look again at the scatterplot from part (a). Which observation is potentially influential? Explain the reason for your choice. (Hint: See Example 4.9.)

The observation (270, 185) is potentially influential because that point has an x value that is far away from the rest of the data set.The observation (300, 145) is potentially influential because that point has an x value that is far away from the rest of the data set. The observation (300, 145) is potentially influential because that point has a large residual.The observation (150, 155) is potentially influential because that point has a large residual.The observation (150, 155) is potentially influential because that point has an x value that is far away from the rest of the data set.

(d)

When the potentially influential observation is deleted from the data set, the equation of the least squares regression line fit to the remaining five observations is

ŷ = −42 + 0.83x.

Use this equation to predict acrylamide concentration (in micrograms per kg) for a frying time of 270 seconds.

micrograms per kg

Compare this prediction to the prediction made in part (b).

Accepted Answer

x = frying time (in seconds) and y = acrylamide concentration (in micrograms per kg) in french…