Question
Find the equation for the least squares regression line of the data described below.
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
Round your answers to the nearest thousandth.
y=
x+
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
Round your answers to the nearest thousandth.
y=
x+
Answers
GPT-4o mini
To find the least squares regression line for the given data, we need to calculate the slope (m) and the y-intercept (b) of the line. The equation of the regression line can be expressed as:
\[ y = mx + b \]
### Step 1: Calculate the necessary sums and means
Given data points:
\[
\begin{align*}
(x_1, y_1) & = (40, 4) \\
(x_2, y_2) & = (55, 7) \\
(x_3, y_3) & = (57, 5) \\
(x_4, y_4) & = (62, 10) \\
(x_5, y_5) & = (63, 9) \\
\end{align*}
\]
Calculate the following sums:
\[
\begin{align*}
n & = 5 \\
\sum x & = 40 + 55 + 57 + 62 + 63 = 277 \\
\sum y & = 4 + 7 + 5 + 10 + 9 = 35 \\
\sum xy & = (40 \cdot 4) + (55 \cdot 7) + (57 \cdot 5) + (62 \cdot 10) + (63 \cdot 9) \\
& = 160 + 385 + 285 + 620 + 567 = 2017 \\
\sum x^2 & = (40^2) + (55^2) + (57^2) + (62^2) + (63^2) \\
& = 1600 + 3025 + 3249 + 3844 + 3969 = 15687 \\
\end{align*}
\]
### Step 2: Calculate the slope (m) and intercept (b)
The formulas for the slope \(m\) and intercept \(b\) are:
\[
m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}
\]
\[
b = \frac{\sum y - m \sum x}{n}
\]
Plug in the values:
\[
m = \frac{5(2017) - (277)(35)}{5(15687) - (277)^2}
\]
Calculating:
1. \( 5(2017) = 10085 \)
2. \( (277)(35) = 9695 \)
3. \( 5(15687) = 78435 \)
4. \( (277)^2 = 76729 \)
Now substitute:
\[
m = \frac{10085 - 9695}{78435 - 76729} = \frac{390}{1716} \approx 0.227
\]
Now for \(b\):
\[
b = \frac{35 - 0.227 \times 277}{5}
\]
Calculating \(0.227 \times 277\):
\[
0.227 \times 277 \approx 62.879
\]
Then:
\[
b = \frac{35 - 62.879}{5} = \frac{-27.879}{5} \approx -5.576
\]
### Final Equation
Putting it all together, the equation of the least squares regression line is:
\[
y \approx 0.227x - 5.576
\]
Rounding to the nearest thousandth:
\[
y = 0.227x - 5.576
\]
\[ y = mx + b \]
### Step 1: Calculate the necessary sums and means
Given data points:
\[
\begin{align*}
(x_1, y_1) & = (40, 4) \\
(x_2, y_2) & = (55, 7) \\
(x_3, y_3) & = (57, 5) \\
(x_4, y_4) & = (62, 10) \\
(x_5, y_5) & = (63, 9) \\
\end{align*}
\]
Calculate the following sums:
\[
\begin{align*}
n & = 5 \\
\sum x & = 40 + 55 + 57 + 62 + 63 = 277 \\
\sum y & = 4 + 7 + 5 + 10 + 9 = 35 \\
\sum xy & = (40 \cdot 4) + (55 \cdot 7) + (57 \cdot 5) + (62 \cdot 10) + (63 \cdot 9) \\
& = 160 + 385 + 285 + 620 + 567 = 2017 \\
\sum x^2 & = (40^2) + (55^2) + (57^2) + (62^2) + (63^2) \\
& = 1600 + 3025 + 3249 + 3844 + 3969 = 15687 \\
\end{align*}
\]
### Step 2: Calculate the slope (m) and intercept (b)
The formulas for the slope \(m\) and intercept \(b\) are:
\[
m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}
\]
\[
b = \frac{\sum y - m \sum x}{n}
\]
Plug in the values:
\[
m = \frac{5(2017) - (277)(35)}{5(15687) - (277)^2}
\]
Calculating:
1. \( 5(2017) = 10085 \)
2. \( (277)(35) = 9695 \)
3. \( 5(15687) = 78435 \)
4. \( (277)^2 = 76729 \)
Now substitute:
\[
m = \frac{10085 - 9695}{78435 - 76729} = \frac{390}{1716} \approx 0.227
\]
Now for \(b\):
\[
b = \frac{35 - 0.227 \times 277}{5}
\]
Calculating \(0.227 \times 277\):
\[
0.227 \times 277 \approx 62.879
\]
Then:
\[
b = \frac{35 - 62.879}{5} = \frac{-27.879}{5} \approx -5.576
\]
### Final Equation
Putting it all together, the equation of the least squares regression line is:
\[
y \approx 0.227x - 5.576
\]
Rounding to the nearest thousandth:
\[
y = 0.227x - 5.576
\]