Question
Find the equation for the least squares regression line of the data described below.
For a science project, Jill wants to see if a larger body of water has more heat energy than a smaller body of water at the same temperature. She prepared a number of buckets filled with various amounts of water at a fixed temperature and dropped an ice cube of the same size into each one.
Jill then recorded the volume of water in each bucket (in milliliters), x, and the amount of time it took for each ice cube to melt (in minutes), y.
Volume of water Melting time (in minutes)
13 16
40 16
69 6
73 9
75 12
Round your answers to the nearest thousandth.
y=
x+
Answers
GPT-4o mini
To find the least squares regression line \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, we can follow these steps:
Given the data:
\[
\begin{array}{|c|c|}
\hline
\text{Volume of water (x)} & \text{Melting time (y)} \\
\hline
13 & 16 \\
40 & 16 \\
69 & 6 \\
73 & 9 \\
75 & 12 \\
\hline
\end{array}
\]
We first calculate the necessary sums:
1. \( n = 5 \) (number of data points)
2. \( \sum x = 13 + 40 + 69 + 73 + 75 = 270 \)
3. \( \sum y = 16 + 16 + 6 + 9 + 12 = 59 \)
4. \( \sum xy = (13 \cdot 16) + (40 \cdot 16) + (69 \cdot 6) + (73 \cdot 9) + (75 \cdot 12) \)
- \( = 208 + 640 + 414 + 657 + 900 = 2829 \)
5. \( \sum x^2 = 13^2 + 40^2 + 69^2 + 73^2 + 75^2 \)
- \( = 169 + 1600 + 4761 + 5329 + 5625 = 11884 \)
Next, we compute the slope \( m \) and the y-intercept \( b \):
The formulas for the slope \( m \) and intercept \( b \) are:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
\[
b = \frac{\sum y - m \sum x}{n}
\]
Now let's calculate \( m \):
\[
m = \frac{5(2829) - (270)(59)}{5(11884) - (270)^2}
\]
Calculating each part:
1. \( 5 \cdot 2829 = 14145 \)
2. \( 270 \cdot 59 = 15930 \)
3. \( 5 \cdot 11884 = 59420 \)
4. \( 270^2 = 72900 \)
Substituting these values into the formula for \( m \):
\[
m = \frac{14145 - 15930}{59420 - 72900}
\]
\[
m = \frac{-1785}{-13480} \approx 0.1324
\]
Now to find the y-intercept \( b \):
\[
b = \frac{59 - (0.1324)(270)}{5}
\]
Calculating:
1. \( 0.1324 \cdot 270 \approx 35.028 \)
2. Now plug this back into \( b \):
\[
b = \frac{59 - 35.028}{5} \approx \frac{23.972}{5} \approx 4.7944
\]
Finally, rounding to the nearest thousandth, we have:
\[
m \approx 0.132 \quad \text{and} \quad b \approx 4.794
\]
Thus, the least squares regression line is:
\[
y = 0.132x + 4.794
\]
Given the data:
\[
\begin{array}{|c|c|}
\hline
\text{Volume of water (x)} & \text{Melting time (y)} \\
\hline
13 & 16 \\
40 & 16 \\
69 & 6 \\
73 & 9 \\
75 & 12 \\
\hline
\end{array}
\]
We first calculate the necessary sums:
1. \( n = 5 \) (number of data points)
2. \( \sum x = 13 + 40 + 69 + 73 + 75 = 270 \)
3. \( \sum y = 16 + 16 + 6 + 9 + 12 = 59 \)
4. \( \sum xy = (13 \cdot 16) + (40 \cdot 16) + (69 \cdot 6) + (73 \cdot 9) + (75 \cdot 12) \)
- \( = 208 + 640 + 414 + 657 + 900 = 2829 \)
5. \( \sum x^2 = 13^2 + 40^2 + 69^2 + 73^2 + 75^2 \)
- \( = 169 + 1600 + 4761 + 5329 + 5625 = 11884 \)
Next, we compute the slope \( m \) and the y-intercept \( b \):
The formulas for the slope \( m \) and intercept \( b \) are:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
\[
b = \frac{\sum y - m \sum x}{n}
\]
Now let's calculate \( m \):
\[
m = \frac{5(2829) - (270)(59)}{5(11884) - (270)^2}
\]
Calculating each part:
1. \( 5 \cdot 2829 = 14145 \)
2. \( 270 \cdot 59 = 15930 \)
3. \( 5 \cdot 11884 = 59420 \)
4. \( 270^2 = 72900 \)
Substituting these values into the formula for \( m \):
\[
m = \frac{14145 - 15930}{59420 - 72900}
\]
\[
m = \frac{-1785}{-13480} \approx 0.1324
\]
Now to find the y-intercept \( b \):
\[
b = \frac{59 - (0.1324)(270)}{5}
\]
Calculating:
1. \( 0.1324 \cdot 270 \approx 35.028 \)
2. Now plug this back into \( b \):
\[
b = \frac{59 - 35.028}{5} \approx \frac{23.972}{5} \approx 4.7944
\]
Finally, rounding to the nearest thousandth, we have:
\[
m \approx 0.132 \quad \text{and} \quad b \approx 4.794
\]
Thus, the least squares regression line is:
\[
y = 0.132x + 4.794
\]