Question
Use the least squares regression line of this data set to predict a value.
Piquant Candy Company is testing a new additive designed to make its hard caramel candies dissolve more slowly. Company scientists gave several taste testers pieces of candy with varying amounts of the additive.
The scientists recorded the amount of additive in each piece (in milligrams), x, and how long the taste tester said it took to dissolve completely (in minutes), y.
Milligrams Minutes
40 4
55 7
57 5
62 10
63 9
The least squares regression line of this data set is:
y=0.229x–5.665
If a piece of candy has 48 milligrams of the additive, how long does the line predict it would take to dissolve completely?
Round your answer to the nearest integer.
minutes
Piquant Candy Company is testing a new additive designed to make its hard caramel candies dissolve more slowly. Company scientists gave several taste testers pieces of candy with varying amounts of the additive.
The scientists recorded the amount of additive in each piece (in milligrams), x, and how long the taste tester said it took to dissolve completely (in minutes), y.
Milligrams Minutes
40 4
55 7
57 5
62 10
63 9
The least squares regression line of this data set is:
y=0.229x–5.665
If a piece of candy has 48 milligrams of the additive, how long does the line predict it would take to dissolve completely?
Round your answer to the nearest integer.
minutes
Answers
Answered by
GPT-4o mini
To predict the dissolve time for a piece of candy with 48 milligrams of the additive using the least squares regression line \( y = 0.229x - 5.665 \), we need to substitute \( x \) with 48.
Plugging in the value:
\[
y = 0.229(48) - 5.665
\]
Calculating \( 0.229 \times 48 \):
\[
0.229 \times 48 = 10.992
\]
Now, substituting back into the equation:
\[
y = 10.992 - 5.665
\]
Calculating that gives:
\[
y = 5.327
\]
Rounding to the nearest integer, we find:
\[
y \approx 5
\]
Thus, the least squares regression line predicts that it would take approximately **5 minutes** for a piece of candy with 48 milligrams of the additive to dissolve completely.
Plugging in the value:
\[
y = 0.229(48) - 5.665
\]
Calculating \( 0.229 \times 48 \):
\[
0.229 \times 48 = 10.992
\]
Now, substituting back into the equation:
\[
y = 10.992 - 5.665
\]
Calculating that gives:
\[
y = 5.327
\]
Rounding to the nearest integer, we find:
\[
y \approx 5
\]
Thus, the least squares regression line predicts that it would take approximately **5 minutes** for a piece of candy with 48 milligrams of the additive to dissolve completely.